## Abstract

There is growing interest among researchers, policy makers, and practitioners in identifying teachers who are skilled at improving student outcomes beyond test scores. However, questions remain about the validity of these teacher effect estimates. Leveraging the random assignment of teachers to classes, I find that teachers have causal effects on their students’ self-reported behavior in class, self-efficacy in math, and happiness in class that are similar in magnitude to effects on math test scores. Weak correlations between teacher effects on different student outcomes indicate that these measures capture unique skills that teachers bring to the classroom. Teacher effects calculated in nonexperimental data are related to these same outcomes following random assignment, revealing that they contain important information content on teachers. However, for some nonexperimental teacher effect estimates, large and potentially important degrees of bias remain. These results suggest that researchers and policy makers should proceed with caution when using these measures. They likely are more appropriate for low-stakes decisions—such as matching teachers to professional development—than for high-stakes personnel decisions and accountability.

## 1. Introduction

Decades’ worth of research on education production have narrowed in on the importance of teachers to student outcomes (Murnane and Phillips 1981; Todd and Wolpin 2003). Over the last several years, these studies have coalesced around two key findings. First, teachers vary considerably in their abilities to improve students’ academic performance (Nye, Konstantopoulos, and Hedges 2004; Hanushek and Rivkin 2010), which in turn influences a variety of long-term outcomes including teenage pregnancy rates, college attendance, and earnings in adulthood (Chetty, Friedman, and Rockoff 2014b). Second, experimental and quasi-experimental studies indicate that “value-added” approaches to estimating teachers’ contributions to student test scores are valid ways to identify effective teachers (Kane and Staiger 2008; Kane et al. 2013; Chetty, Friedman, and Rockoff 2014a; Glazerman and Protik 2015; Bacher-Hicks et al. 2017). In other words, on average, these teacher effect estimates are not confounded with the nonrandom sorting of teachers to students, the specific set of students in the classroom, or factors beyond teachers’ control. Policy makers have taken notice of these findings, leading to widespread changes in teacher evaluation, compensation, and promotion.

While these studies have focused predominantly on teachers’ impact on students’ academic performance, the research community is starting to collect evidence that teachers also vary in their contributions to a variety of other student outcomes in ways that are only weakly related to their effects on test scores (Jennings and DiPrete 2010; Jackson 2012; Gershenson 2016; Kraft, forthcoming). For example, in work drawing on the study that generated data used in this paper, Blazar and Kraft (2017) found that teachers identified as 1 standard deviation (SD) above the mean in the distribution of effectiveness improved students’ self-reported behavior in class, self-efficacy in math, and happiness in class by between 0.15 SD to 0.30 SD. These effects are similar to or larger than teacher effects on students’ test scores (Hanushek and Rivkin 2010). However, teachers who were effective at improving these outcomes often were not equally effective at improving students’ math test scores, with correlations between teacher effect estimates no higher than 0.19. Jackson (2012) came to similar conclusions using additional student outcomes, and also found that teacher effects on non-tested outcomes captured in ninth grade predicted longer-run outcomes, including high school completion above and beyond teachers’ effects on test scores (Jackson 2016). Together, these findings lend empirical evidence to the multidimensional nature of teaching and, thus, the need for policy makers to account for this sort of complexity.

Given that the research base examining teachers’ contributions to student outcomes beyond test scores is relatively new, important questions remain about the validity of these measures. In the value-added literature more broadly, researchers have asked about the sensitivity of teacher effects to different model specifications and the specific set of covariates included in the model (Goldhaber and Theobald 2012), as well as the most appropriate ways to calculate these scores in light of measurement error (Guarino et al. 2015). Further, it is not clear whether the key identifying assumption underlying the estimation of teacher effects—that estimates are not biased by nonrandom sorting of students to teachers (Kane et al. 2013; Chetty, Friedman, and Rockoff 2014a)—holds when test scores are replaced with other student outcomes. Researchers who estimate value added to students’ test scores typically control for prior test scores because they capture many of the predetermined factors that also affect current achievement, including the schools students attend, their neighborhoods, and the family members with whom they interact. However, it is possible there are additional factors not captured by prior test scores or by prior measures of the outcome variable that lead to bias in teacher effects on other student outcomes beyond test scores.

I examine these issues by drawing on a dataset in which participating students completed a survey that asked about a range of attitudes and behaviors in class. In the third year of the study, a subset of participating teachers (*N* = 41) was randomly assigned to class rosters within schools. Together, these data allow me to examine the extent to which teachers vary in their contribution to students’ attitudes and behaviors, even after random assignment; the sensitivity of teacher effects on students’ attitudes and behaviors to different model specifications, including those that control for students’ prior academic performance versus prior attitudes and behaviors; and, ultimately, whether nonexperimental estimates of teacher effects on these attitudes and behaviors predict these same outcomes following random assignment, which produces a measure of forecast bias.

Findings indicate that teachers have causal effects on students’ self-reported behavior in class, self-efficacy in math, and happiness in class. The magnitude of the teacher-level variation on these outcomes is similar to or larger than effects on math test scores (e.g., Hanushek and Rivkin 2010). Weak correlations between teacher effects on different student outcomes indicate that these measures capture unique skills that teachers bring to the classroom. However, in some cases, value-added approaches to estimating these teacher effects appear to be insufficient to account for all sources of bias. One exception is teacher effects on students’ behavior in class, where predicted differences perfectly predict actual differences following random assignment. In the observational portion of this study, teacher effects are not particularly sensitive to models that control for students’ prior achievement, student demographic characteristics, or prior survey responses. Given that these are the tools and data typically available to the econometrician, it is not clear that bias could easily be reduced further. In turn, it will be important for researchers and policy makers to use these estimates of teacher effectiveness with caution. In the Conclusion, I describe some potential uses of these measures, focusing on low-stakes decision making (such as matching teachers to professional development) rather than high-stakes decisions (such as teacher evaluation and promotion).

## 2. Validating Methods for Estimating Teacher Effects on Student Outcomes

Over the last decade, several experimental and quasi-experimental studies have tested the validity of nonexperimental methods for estimating teacher effects on student achievement. In the first of these, Kane and Staiger (2008) described the rationale and setup for such a study: “Non-experimental estimates of teacher effects attempt to answer a very specific question: If a given classroom of students were to have teacher A rather than teacher B, how much different would their average test scores be at the end of the year?” (p. 1). However, as these sorts of teacher effects estimates are derived from conditions where nonrandom sorting is the norm (Clotfelter, Ladd, and Vigdor 2006; Rothstein 2010), these models assume that statistical controls (e.g., students’ prior achievement, demographic characteristics) are sufficient to isolate the talents and skills of individual teachers rather than “principals’ preferential treatment of their favorite colleagues, ability-tracking based on information not captured by prior test scores, or the advocacy of engaged parents for specific teachers” (Kane and Staiger 2008, p. 1).^{1}

Random assignment of teachers to classes offers a way to test this assumption. If nonexperimental teacher effects are causal estimates that capture true differences in quality between teachers, then nonexperimental or predicted differences should be equal, on average, to actual differences following the random assignment of teachers to classes. In other words, a 1 SD increase in predicted differences in achievement across classrooms should result in a 1 SD increase in observed differences, on average. Estimates that are statistically significantly greater than 0 SD indicate that nonexperimental teacher effects contain some information content about teachers’ underlying talents and skills. However, deviations from the 1:1 relationship would signal that these scores also are influenced by factors beyond teachers’ control, including students’ background and skill, the composition of students in the classroom, or strategic assignment policies. These deviations often are referred to as “forecast bias.”

Results from Kane and Staiger (2008) and other experimental studies (Kane et al. 2013; Glazerman and Protik 2015; Bacher-Hicks et al. 2017) have accumulated to provide strong evidence against bias in teacher effects on students’ test scores. Pooling results from three experimental studies with the same research design (i.e., teachers randomly assigned to class rosters within schools^{2}), Bacher-Hicks et al. (2017) found an estimate of 0.96 SD relating predicted, nonexperimental teacher effects on students’ math achievement to actual differences in this same outcome following random assignment. Predicted teacher effects were calculated from models that controlled for students’ prior achievement. Given the nature of their meta-analytic approach, the standard error around this estimate (0.099) was much smaller than in each individual study, and the corresponding 95 percent confidence interval included 1 SD, indicating little or no bias. This result was quite similar to findings from quasi-experimental studies in much larger administrative datasets, which leveraged plausibly exogenous variation in teacher assignments due to staffing changes at the school-grade level (Bacher-Hicks, Kane, and Staiger 2014; Chetty, Friedman, and Rockoff 2014a).

Following a long line of inquiry around the sensitivity of value-added estimates to different model specifications and which may be most appropriate for policy (Aaronson, Barrow, and Sander 2007; Newton et al. 2010; Goldhaber and Theobald 2012; Blazar, Litke, and Barmore 2016), many of these studies also examined the predictive validity of alternative methods for estimating teacher effects. For example, some have advocated for controlling for the composition of students in the classroom, which is thought to influence test scores beyond teachers themselves (Hanushek et al. 2003; Kupermintz 2003). Others have specified models that only compare teachers within schools in order to limit bias due to sorting of teachers and students across schools (Rivkin, Hanushek, and Kain 2005); however, this approach can lead to large differences in teacher rankings relative to models that compare teachers across schools (Goldhaber and Theobald 2012). The general conclusion across validation studies is that controlling for students’ prior achievement is sufficient to account for the vast majority of bias in teacher effect estimates on achievement (Chetty, Friedman, and Rockoff 2014a; Kane et al. 2013; Kane and Staiger 2008).

To my knowledge, only one study has examined the predictive validity of teacher effects on student outcomes beyond test scores.^{3} Drawing on the quasi-experimental design described by Chetty, Friedman, and Rockoff (2014a), Backes and Hansen (2018) examined the validity of teacher effects on a range of observed school behaviors captured in administrative records. They found that teacher effects on students’ suspensions and percent of classes failed did not contain bias when pooling across all grade levels. However, teacher effects on unexcused absences, grade point average, and on-time grade progression did contain moderate to large degrees of bias, as least in some grade levels. For teacher effects on both unexcused absences and on-time grade progression, predicted differences at the elementary level overstated actual differences (i.e., coefficient less than 1 SD), likely due to sorting of higher-performing students to higher-performing teachers in a way that could not be controlled for in the model. The opposite was true at the high school level, where predicted differences in teacher effectiveness understated actual differences (i.e., coefficient greater than 1 SD). This suggests that bias in teacher effects on outcomes beyond test scores may not be easily quantified or classified across contexts.

## 3. Data and Sample

As in Blazar and Kraft (2017) and Bacher-Hicks et al. (2017), this paper draws on data from the National Center for Teacher Effectiveness (NCTE), whose goal was to develop valid measures of effective teaching in upper-elementary mathematics. Over the course of three school years (2010–11 through 2012–13), the project collected data from participating fourth- and fifth-grade teachers (*N* = 310) in four anonymous districts from three states on the east coast of the United States. Participants were generalists who taught all subject areas. This is important, as it provided an opportunity to estimate the contribution of individual teachers to students’ attitudes and behaviors that was not confounded with the effect of another teacher with whom a student engaged in the same year. Teacher–student links were verified for all study participants based on class rosters provided by teachers.

Measures of students’ attitudes and behaviors came from a survey administered in the spring of each school year (see Appendix table A.1 for survey item text and descriptive statistics). Based on theory and exploratory factor analyses (see Blazar and Kraft 2017), I divided items into three constructs: *Behavior in Class* (internal consistency reliability [$\alpha $] is 0.74), *Self-Efficacy in Math* ($\alpha $ = 0.76), and *Happiness in Class* ($\alpha $ = 0.82). Teacher reports of student behavior and self-reports of versions of the latter two constructs have been linked to labor market outcomes even controlling for cognitive ability (Lyubomirsky, King, and Diener 2005; Mueller and Plug 2006; Chetty et al. 2011), lending strong consequential validity to these metrics. Blazar and Kraft (2017) describe additional validity evidence, including convergent validity, for these constructs. For each of these outcomes, I created final scales by reverse coding items with negative valence, averaging student responses across all available items, and then standardizing to mean of 0 and SD of 1.^{4} Standardization occurred within school year but across grades.

Student demographic and achievement data came from district administrative records. Demographic data included sex, race/ethnicity, free- or reduced-price lunch (FRPL) eligibility, limited English proficiency (LEP) status, and special education (SPED) status. These records also included current- and prior-year test scores in math and reading on state assessments, which were standardized within district by grade, subject, and year using the entire population of students in each district, grade, subject, and year.

I focus on two subsamples from the larger group of 310 teachers. The primary analytic sample includes the subset of 41 teachers who were part of the random assignment portion of the NCTE study in the third year of data collection. I describe this sample and the experimental design in the next section. The second sample includes the set of students (and their teachers) who took the project-administered survey in both the current and prior years. This allowed me to test the sensitivity of teacher effect estimates to different model specifications, including those that controlled for students’ prior survey responses, from a balanced sample of teachers and students. As noted previously, the student survey only was administered in the spring of each year; therefore, this sample consisted of the group of fifth-grade teachers who happened to have students who also were part of the NCTE study in the fourth grade (*N* = 51 teachers; *N* = 548 students).^{5}

Generally, I found that average teacher characteristics, including their sex, race, math course taking, math knowledge, route to certification, years of teaching experience, and value-added scores calculated from state math tests were similar across samples (see table 1).^{6} Given that teachers self-selected into the NCTE study, I also tested whether these samples differed from the full population of fourth- and fifth-grade teachers in each district with regard to value-added scores on the state math test (see equation 1 for more details on these value-added predictions). Although I found a marginally significant difference between the full NCTE sample and the district populations (*p* = 0.065), I found no difference between the district populations and either the experimental or nonexperimental subsamples used in this analysis (*p* = 0.890 and 0.652, respectively; not shown in table 1). These similarities lend external validity to my findings.

. | . | Experimental Sample . | Nonexperimental Sample . | District Populations . | |||
---|---|---|---|---|---|---|---|

. | Full NCTE Sample . | Mean . | p-value on Difference
. | Mean . | p-value on Difference
. | Mean . | p-value on Difference
. |

Male | 0.16 | 0.15 | 0.95 | 0.19 | 0.604 | — | — |

African American | 0.22 | 0.18 | 0.529 | 0.24 | 0.790 | — | — |

Asian | 0.03 | 0.05 | 0.408 | 0.00 | 0.241 | — | — |

Hispanic | 0.03 | 0.03 | 0.866 | 0.02 | 0.686 | — | — |

White | 0.65 | 0.70 | 0.525 | 0.67 | 0.807 | — | — |

Mathematics coursework | 2.58 | 2.62 | 0.697 | 2.54 | 0.735 | — | — |

Mathematical content knowledge | 0.01 | 0.05 | 0.816 | 0.07 | 0.671 | — | — |

Alternative certification | 0.08 | 0.08 | 0.923 | 0.12 | 0.362 | — | — |

Teaching experience | 11.04 | 14.35 | 0.005 | 11.44 | 0.704 | — | — |

Value added on state math test | 0.02 | 0.00 | 0.646 | 0.01 | 0.810 | 0.00 | 0.065 |

p-value on joint test | 0.533 | 0.958 | NA | ||||

Teachers | 310 | 41 | 51 | 3,454 |

. | . | Experimental Sample . | Nonexperimental Sample . | District Populations . | |||
---|---|---|---|---|---|---|---|

. | Full NCTE Sample . | Mean . | p-value on Difference
. | Mean . | p-value on Difference
. | Mean . | p-value on Difference
. |

Male | 0.16 | 0.15 | 0.95 | 0.19 | 0.604 | — | — |

African American | 0.22 | 0.18 | 0.529 | 0.24 | 0.790 | — | — |

Asian | 0.03 | 0.05 | 0.408 | 0.00 | 0.241 | — | — |

Hispanic | 0.03 | 0.03 | 0.866 | 0.02 | 0.686 | — | — |

White | 0.65 | 0.70 | 0.525 | 0.67 | 0.807 | — | — |

Mathematics coursework | 2.58 | 2.62 | 0.697 | 2.54 | 0.735 | — | — |

Mathematical content knowledge | 0.01 | 0.05 | 0.816 | 0.07 | 0.671 | — | — |

Alternative certification | 0.08 | 0.08 | 0.923 | 0.12 | 0.362 | — | — |

Teaching experience | 11.04 | 14.35 | 0.005 | 11.44 | 0.704 | — | — |

Value added on state math test | 0.02 | 0.00 | 0.646 | 0.01 | 0.810 | 0.00 | 0.065 |

p-value on joint test | 0.533 | 0.958 | NA | ||||

Teachers | 310 | 41 | 51 | 3,454 |

*Note:**p*-value refers to difference from the full National Center for Teacher Effectiveness (NCTE) sample.

## 4. Experimental Design

In the spring of 2012, the NCTE project team worked with staff at participating schools to randomly assign sets of teachers to class rosters of the same grade level (i.e., fourth or fifth grade) that were constructed by principals or school leaders. To be eligible for randomization, teachers had to work in schools and grades in which there was at least one other participating teacher. In addition, their principal had to consider these teachers as capable of teaching any of the rosters of students designated for the group of teachers.

In order to fully leverage this experimental design, it was important to limit the most pertinent threats to internal validity: attrition and noncompliance among participating teachers and students (Murnane and Willett 2011). My general approach was to focus on randomization blocks in which attrition and noncompliance were not a concern. As these blocks are analogous to individual experiments, dropping individual ones should not threaten the internal validity of results. First, I restricted the sample to blocks where teachers and their randomization block partner(s) had both current-year student outcomes and prior-year, nonexperimental teacher effect estimates. Of the original 79 teachers who agreed to participate and were randomly assigned to class rosters within schools^{7}, I dropped seven teachers who left the study before the beginning of the 2012–13 school year for reasons unrelated to the experiment (i.e., leaving the district or teaching, maternity leave, change in teaching assignment); eleven teachers who only were part of the study in the third year and, therefore, did not have the necessary data from prior years to calculate nonexperimental teacher effects on students’ attitudes and behaviors; and seven teachers whose random assignment partner(s) left the study for either of the two reasons above.^{8}

Next, I restricted the remaining sample to randomization blocks with low levels of noncompliance among participating students. Here, noncompliance refers to the fact that some students switched out of their randomly assigned teacher's classroom. Other studies that exploit random assignment between teachers and students have accounted for this form of noncompliance through instrumental variables estimation and calculation of treatment on the treated (Kane et al. 2013; Glazerman and Protik 2015; Bacher-Hicks et al. 2017). However, this approach was not possible in this study, given that students who transferred out of an NCTE teacher's classroom no longer had survey data to calculate teacher effects on these outcomes. Further, I would have needed to have prior student survey responses for these students’ actual teachers, which I did not. In total, 28 percent of students moved out of their randomly assigned teachers’ classroom (see Appendix table A.2 for information on reasons for and patterns of noncompliance). At the same time, noncompliance was nested within a small subset of six randomization blocks. In these blocks, rates of noncompliance ranged from 40 percent to 82 percent due primarily to principals and school leaders who made changes to the originally constructed class rosters. By eliminating these blocks, I am able to focus on a sample with a much lower rate of noncompliance (11 percent) and where patterns of noncompliance are much more typical. The remaining eighteen blocks had a total of 67 noncompliers and an average rate of noncompliance of 9 percent per block; three randomization blocks had full compliance.

In table 2, I confirm the success of the randomization process among the teachers in my final analytic sample (*N* = 41) and the students on their randomly assigned rosters (*N* = 598).^{9} In a traditional experiment, one can examine balance at baseline by calculating differences in average student characteristics between the treatment and control groups. In this context, though, treatment consisted of multiple possible teachers within a given randomization block. Thus, to examine balance, I examined the relationship between the assigned teacher's predicted effectiveness at improving students’ state math test scores in years prior to the experiment and baseline student characteristics. Specifically, I regressed these teacher effect estimates on a vector of observable student characteristics and fixed effects for randomization block. As expected, observable student characteristics were not related to teacher effects on state math tests, either tested individually or as a group (*p* = 0.808), supporting the fidelity of the randomization process. Students who did not stay in their randomly assigned teacher's classroom (i.e., “noncompliers”) look similar to compliers at improving state math test scores in years prior to random assignment, based on observable baseline characteristics, as well as the observed effectiveness of their randomly assigned teacher (see Appendix table A.3).^{10} As such, I am less concerned about having to drop the few noncompliers left in my sample from all subsequent analyses.

. | Teacher Effects on State Math Scores from Randomly Assigned Teacher . |
---|---|

Male | −0.005 |

(0.009) | |

African American | 0.028 |

(0.027) | |

Asian | 0.030 |

(0.029) | |

Hispanic | 0.043 |

(0.028) | |

White | 0.010 |

(0.028) | |

FRPL | 0.002 |

(0.011) | |

SPED | −0.023 |

(0.021) | |

LEP | 0.004 |

(0.014) | |

Prior achievement on state math test | 0.009 |

(0.007) | |

Prior achievement on state reading test | −0.001 |

(0.007) | |

p-value on joint test | 0.316 |

Teachers | 41 |

Students | 598 |

. | Teacher Effects on State Math Scores from Randomly Assigned Teacher . |
---|---|

Male | −0.005 |

(0.009) | |

African American | 0.028 |

(0.027) | |

Asian | 0.030 |

(0.029) | |

Hispanic | 0.043 |

(0.028) | |

White | 0.010 |

(0.028) | |

FRPL | 0.002 |

(0.011) | |

SPED | −0.023 |

(0.021) | |

LEP | 0.004 |

(0.014) | |

Prior achievement on state math test | 0.009 |

(0.007) | |

Prior achievement on state reading test | −0.001 |

(0.007) | |

p-value on joint test | 0.316 |

Teachers | 41 |

Students | 598 |

*Notes:* The regression model includes fixed effects for randomization block. Robust standard errors in parentheses. FRPL = free- or reduced-price lunch eligible; LEP = limited English proficiency status; SPED = special education status.

## 5. Empirical Strategy

*Behavior in Class, Self-Efficacy in Math,*and

*Happiness in Class*—for student

*i*in district

*d*, school

*s*, grade

*g*taught by teacher

*j*in year

*t*. As a point of comparison, I also specify models that use students’ math achievement as an outcome. Throughout the paper, I test a variety of alternative models that include different combinations of control variables. The full set of controls includes a cubic function of students’ prior academic achievement, $Ait-1$, in both math and reading; a prior measure of the outcome variable, $OUTCOMEit-1$; student demographic characteristics, $Xit$, including gender, race, FRPL eligibility, SPED status, and LEP status; these same test-score variables and demographic characteristics averaged to the class level, $X\u203eitc$, and to the school level, $X\u203eits$; and school fixed effects, $\sigma s$, which replace school characteristics in some models.

To generate teacher effect estimates, which I refer to as $\tau ^jtS$, I took two approaches, each with strengths and limitations. First, I calculated teacher effects by fitting equation 1 using ordinary least squares (OLS) regression and then averaging student-level residuals to the teacher level. I did so separately for each outcome measure, as well as with several different model specifications denoted by the superscript, *S*. This approach is intuitive, as it creates an estimate of the contribution of teachers to student outcomes above and beyond factors already controlled for in the model. It also is computationally simple.^{11} At the same time, measurement error in these estimates due to, for example, small class sizes, sampling idiosyncrasies, and measurement error in students’ survey responses may lead me to overstate the variance of true teacher effects; it also could attenuate the relationship between different measures of teacher effectiveness (e.g., measures at two points in time), even if they capture the same underlying construct.

Therefore, I also calculated a form of empirical Bayes estimates that take into account measurement error and shrink teacher effects back toward the mean based on their precision. To do so, I included a teacher-level random effect in the model, which I fit using restricted maximum likelihood. This approach is similar to a two-step approach described by others (e.g., Kane et al. 2013; Chetty, Friedman, and Rockoff 2014a; Guarino et al. 2015) that first calculates the unshrunken teacher effects using OLS and then multiplies these by a shrinkage factor. The shrinkage factor generally is calculated by looking at variation in teacher effects within teachers and across classrooms. It was not possible to use this two-step approach here given that data from multiple classrooms from the same teacher were not available in the experimental portion of the study; elementary teachers in the sample all worked with just one class in a given year. Instead, the one-step random effects approach I use shrinks estimates back toward the mean based on the variance of the observed data (Raudenbush and Bryk 2002). Although shrinking teacher effects is commonplace in both research and policy (Koedel, Mihaly, and Rockoff 2015), theory and simulated analyses show that shrunken estimates are biased downward relative to the size of the measurement error (Jacob and Lefgren 2005). I refer to these two sets of estimates as “unshrunken” and “shrunken” teacher effects.

I utilized these teacher effect estimates for three subsequent analyses. First, I estimated the variance of $\tau ^jtS$ in order to examine the extent to which teachers vary in their contributions to students’ attitudes and behaviors. I focused on the experimental sample in order to be assured that estimates were not biased by nonrandom sorting. Given that the variance of true teacher effects is bounded between the unshrunken and shrunken estimates (Raudenbush and Bryk 2002), I present both. The latter are model-based estimates reported directly from the random effects model. By calculating pairwise correlations between these teacher effect estimates, I also examined whether teachers who improved one student outcome were equally effective at improving others.

Second, I examined the sensitivity of $\tau ^jtS$ to different model specifications. I began with a baseline model that calculated teacher effects controlling only for students’ prior academic achievement, as this is the measure typically used to account for nonrandom sorting when test scores are the outcome of interest (Kane and Staiger 2008; Kane et al. 2013; Chetty, Friedman, and Rockoff 2014a). I also considered a model that conditioned estimates on a lagged measure of students’ survey response, which is a more direct analog of the value-added approach by looking at gains in student outcomes. Additional variations of these models include ones that control for student, class, or school characteristics. In order to address concerns about “reference bias” in self-reported survey measures (Duckworth and Yeager 2015; West et al. 2016), I also replaced school characteristics with school fixed effects. By making within-school comparisons, I am able to difference out school-level factors, including norms around behavior or engagement, that can create an implicit standard of comparison students use when judging their own behavior or engagement. It was not possible to run this second set of analyses in the experimental sample, given that only a small subset of students and teachers in that sample had lagged survey measures. There also was no guarantee that a teacher who had students with prior survey measures had a randomization block partner whose students had these measures. Instead, I focused on the balanced sample of teachers and students with all possible control variables from the larger observational dataset.

*i*in teacher

*j*’s classroom in the 2012–13 school year. I predicted these measures in the random assignment year with predicted, nonexperimental teacher effect estimates, $\tau ^jt<2012-13S$. That is, when

*Behavior in Class*is the outcome of interest, $\tau ^jt<2012-13S$ represents a nonexperimental estimate of teachers’ effectiveness at improving students’

*Behavior in Class*in prior years; when

*Self-Efficacy in Math*is the outcome of interest, $\tau ^jt<2012-13S$, represents a nonexperimental estimate of teachers’ effectiveness at improving

*Self-Efficacy in Math*in prior years. Following the research design, I included fixed effects for each randomization block, $\nu sg$. In order to increase the precision of my estimates, I calculated nonexperimental teacher effects using all available teacher-years prior to the experiment. For the same reason, in equation 2, I also controlled for students’ prior achievement, demographic characteristics, and class characteristics captured from the randomly assigned rosters. I clustered standard errors at the class level to account for the nested structure of the data.

My parameter of interest is $\delta $, which describes the relationship between nonexperimental teacher effect estimates and current student outcomes. As in Kane and Staiger (2008), I examined whether these estimates had any predictive validity (i.e., whether they were statistically significantly different from 0 SD) and whether they contained some degree of bias (i.e., whether they were statistically significantly different from 1 SD).

## 6. Results

### Experimental Teacher Effects on Students’ Attitudes and Behaviors

In table 3, I present results describing the extent to which teachers vary in their contributions to students’ attitudes and behaviors, as well as their math achievement. Estimates represent the standard deviation of the teacher-level variance, with panel A and panel B presenting unshrunken and shrunken estimates, respectively. In table 4, I present correlations between corresponding unshrunken and shrunken estimates. All models focus on the experimental sample in which teachers were randomly assigned to class rosters within schools, and therefore include randomization block (i.e., school-by-grade) fixed effects to match this design. Model 1 estimates teacher effects with no additional controls, and model 2 adds students’ prior achievement in math and reading, which is standard practice when estimating teacher effects. In model 3, I add student demographic characteristics, as well as class characteristics that aim to remove the contribution of peer effects from the teacher effect estimates. It was not possible to model classroom-level shocks directly, as random assignment data were not available over multiple classes or school years. Class characteristics describe the set of students included on the randomly assigned rosters rather than the students who ultimately stayed in that classroom.

. | (1) . | (2) . | (3) . |
---|---|---|---|

Panel A: Unshrunken Estimates . | |||

State math test | 0.28 | 0.19 | 0.13 |

Behavior in Class | 0.28 | 0.24 | 0.14 |

Self-Efficacy in Math | 0.29 | 0.27 | 0.19 |

Happiness in Class | 0.35 | 0.35 | 0.26 |

Panel B: Shrunken Estimates | |||

State math test | 0.22 | 0.18 | 0.13 |

Behavior in Class | 0.13 | 0.09 | 0.05 |

Self-Efficacy in Math | 0.00 | 0.00 | 0.08 |

Happiness in Class | 0.33 | 0.33 | 0.34 |

Prior achievement | X | X | |

Student characteristics | X | ||

Class characteristics | X | ||

School-by-grade fixed effects | X | X | X |

Teachers | 41 | 41 | 41 |

Students | 531 | 531 | 531 |

. | (1) . | (2) . | (3) . |
---|---|---|---|

Panel A: Unshrunken Estimates . | |||

State math test | 0.28 | 0.19 | 0.13 |

Behavior in Class | 0.28 | 0.24 | 0.14 |

Self-Efficacy in Math | 0.29 | 0.27 | 0.19 |

Happiness in Class | 0.35 | 0.35 | 0.26 |

Panel B: Shrunken Estimates | |||

State math test | 0.22 | 0.18 | 0.13 |

Behavior in Class | 0.13 | 0.09 | 0.05 |

Self-Efficacy in Math | 0.00 | 0.00 | 0.08 |

Happiness in Class | 0.33 | 0.33 | 0.34 |

Prior achievement | X | X | |

Student characteristics | X | ||

Class characteristics | X | ||

School-by-grade fixed effects | X | X | X |

Teachers | 41 | 41 | 41 |

Students | 531 | 531 | 531 |

. | (1) . | (2) . | (3) . |
---|---|---|---|

Teacher effects on state math test | 0.93 | 0.95 | 0.84 |

Teacher effects on Behavior in Class | 0.87 | 0.84 | 0.87 |

Teacher effects on Self-Efficacy in Math | — | — | 0.83 |

Teacher effects on Happiness in Class | 0.95 | 0.95 | 0.87 |

Prior achievement | X | X | |

Student characteristics | X | ||

Class characteristics | X | ||

School-by-grade fixed effects | X | X | X |

Teachers | 41 | 41 | 41 |

. | (1) . | (2) . | (3) . |
---|---|---|---|

Teacher effects on state math test | 0.93 | 0.95 | 0.84 |

Teacher effects on Behavior in Class | 0.87 | 0.84 | 0.87 |

Teacher effects on Self-Efficacy in Math | — | — | 0.83 |

Teacher effects on Happiness in Class | 0.95 | 0.95 | 0.87 |

Prior achievement | X | X | |

Student characteristics | X | ||

Class characteristics | X | ||

School-by-grade fixed effects | X | X | X |

Teachers | 41 | 41 | 41 |

*Note:* Empty cells indicate that variation in unshrunken or shrunken teacher effects is close to 0 SD (see table 3).

I begin by describing the magnitude of teacher effects on students’ math performance on state tests, which have been well documented in the academic literature (for a review, see Hanushek and Rivkin 2010) and thus provide a point of comparison for the magnitude of teacher effects on students’ attitudes and behaviors. I find that a 1 SD increase in teacher effectiveness is equivalent to between a 0.13 SD and 0.28 SD increase in students’ math achievement. Results are fairly similar between the corresponding unshrunken and shrunken estimates, particularly when controlling for students’ prior achievement. In models 2 and 3, the magnitude of the teacher-level variation for these unshrunken and shrunken estimates are almost identical to two decimal places, and correlations between them range from 0.84 to 0.95. These results are quite similar to those found by Guarino et al. (2015), who argued that the “effect of shrinkage itself does not appear to be practically important for properly ranking teachers or to ameliorate the performance of the [unshrunken] estimator” (p. 212).

Whereas I do not find large differences in the magnitude of teacher effects on math test scores between shrunken and unshrunken estimates, I do observe differences depending on the set of covariates included in the model. In particular, the variance of teacher effects on math test scores is substantively larger in model 1 (0.28 and 0.22 SD for unshrunken and shrunken estimates, respectively), which only controls for randomization block fixed effects, than in model 3 (0.13 SD for both unshrunken and shrunken estimates), which also controls for student and class characteristics. This is consistent with other literature, suggesting that controlling for observable class or peer characteristics produces a conservative estimate of the magnitude of teacher effects on student test scores (Kane et al. 2013; Thompson, Guarino, and Wooldridge 2015). In model 3, both shrunken and unshrunken teacher effect estimates of 0.13 SD indicate that, relative to an average teacher, teachers at the 84th percentile of the distribution of effectiveness move the median student up to roughly the 55th percentile of math achievement.

I also find that teachers have substantive impacts on self-reported measures of students’ attitudes and behaviors. The largest of these teacher effects is on students’ *Happiness in Class*, where a 1 SD increase in teacher effectiveness leads to a roughly 0.30 SD increase in this outcome. Similar to teacher effects on students’ math performance, results for teacher effects on students’ *Happiness in Class* are fairly consistent between panel A and panel B, indicating that shrinkage does not necessarily boost performance. Correlations between unshrunken and shrunken estimates range from 0.87 to 0.95. For the unshrunken teacher effects, estimates are smaller when controlling for student and class characteristics in model 3 (0.26 SD) compared with estimates from the other two models (0.35 SD). This is not the case for the shrunken teacher effects, where estimates across all three models are roughly 0.33 SD. In model 3, the variance of the unshrunken teacher effects on students’ *Happiness in Class* is slightly larger than the variance of the analogous shrunken estimates. This is possible given that, as described above, the shrunken estimates are not derived by directly shrinking the unshrunken estimates. Rather, these estimates come from a separate model that includes a teacher-level random effect and generates model-based estimates of the variance component.

The evidence also points to sizeable teacher effects on students’ *Behavior in Class* and *Self-Efficacy in Math*, though results are less consistent between unshrunken and shrunken estimators. Without shrinkage, the magnitude of teacher effects on these two outcomes generally is larger than teacher effects on students’ math performance but smaller than teacher effects on students’ *Happiness in Class*: between 0.14 SD and 0.28 SD for teacher effects on students’ *Behavior in Class,* and between 0.19 SD and 0.29 SD for teacher effects on students’ *Self-Efficacy in Math*. Shrunken estimates are considerably smaller. For example, in model 3, these shrunken estimates are 0.05 SD for teacher effects on *Behavior in Class* and 0.08 SD for teacher effects on *Self-Efficacy in Math*. Correlations between the unshrunken and shrunken estimates still are strong, but never above 0.87. Models that exclude class characteristics and use shrinkage to calculate teacher effects on students’ *Self-Efficacy in Math* produce estimates close to 0 SD. For this reason, I exclude from table 4 correlations between unshrunken and shrunken estimates for this outcome and these models.

It is counterintuitive that models that include class characteristics produce estimates that are larger than those that exclude these control variables. It is possible the error structure for students’ self-reported *Self-Efficacy in Math* is quite different from the error structure for other measures, which in turn leads to challenges when implementing shrinkage through a random effects model fit using restricted maximum likelihood estimation. Although restricted maximum likelihood aims to address concerns that full maximum likelihood tends to produce variance estimates that are biased downward, this may also be a concern in the relatively small sample of teachers and students (Harville 1977; Raudenbush and Bryk 2002). Mixed models can result in singular fits (i.e., variance-covariance components that are exactly zero) in several instances, including a small number of random effects and complex random effects models (Gelman 2006). This topic is beyond the scope of this paper but is an important one for future research.

In table 5, I present a correlation matrix of teacher effects on different student outcomes. Here, teacher effects come from model 3 (see tables 3 and 4), which controls for prior achievement, student characteristics, and class characteristics. I focus on this model given that the magnitude of shrunken and unshrunken teacher effects are greater than 0 SD for all outcomes. One concern when estimating relationships between different measures of teacher quality is that individual teacher effect estimates are measured with error, which will attenuate these correlations (Spearman 1904). Indeed, correlations between the shrunken teacher effect estimates in panel B—which are estimated in a way that aims to reduce measurement error—generally are larger than correlations between the unshrunken ones in panel A. At the same time, differences in correlations between these two panels are not large, suggesting that additional approaches to address attenuation due to measurement error are unlikely to change overall patterns of results.

. | Teacher Effects on State Math Test . | Teacher Effects on Behavior in Class . | Teacher Effects on Self-Efficacy in Math . | Teacher Effects on Happiness in Class . |
---|---|---|---|---|

Panel A: Unshrunken Estimates . | ||||

Teacher effects on state math test | 1.0 | |||

Teacher effects on Behavior in Class | 0.16 | 1.0 | ||

Teacher effects on Self-Efficacy in Math | 0.17 | 0.48^{**} | 1.0 | |

Teacher effects on Happiness in Class | −0.22 | 0.17 | 0.44^{**} | 1.0 |

Panel B: Shrunken Estimates | ||||

Teacher effects on state math test | 1.0 | |||

Teacher effects on Behavior in Class | 0.17 | 1.0 | ||

Teacher effects on Self-Efficacy in Math | −0.03 | 0.65^{***} | 1.0 | |

Teacher effects on Happiness in Class | −0.38^{*} | 0.17 | 0.59^{***} | 1.0 |

. | Teacher Effects on State Math Test . | Teacher Effects on Behavior in Class . | Teacher Effects on Self-Efficacy in Math . | Teacher Effects on Happiness in Class . |
---|---|---|---|---|

Panel A: Unshrunken Estimates . | ||||

Teacher effects on state math test | 1.0 | |||

Teacher effects on Behavior in Class | 0.16 | 1.0 | ||

Teacher effects on Self-Efficacy in Math | 0.17 | 0.48^{**} | 1.0 | |

Teacher effects on Happiness in Class | −0.22 | 0.17 | 0.44^{**} | 1.0 |

Panel B: Shrunken Estimates | ||||

Teacher effects on state math test | 1.0 | |||

Teacher effects on Behavior in Class | 0.17 | 1.0 | ||

Teacher effects on Self-Efficacy in Math | −0.03 | 0.65^{***} | 1.0 | |

Teacher effects on Happiness in Class | −0.38^{*} | 0.17 | 0.59^{***} | 1.0 |

The largest of these correlations is between teacher effects on different measures of students’ attitudes and behaviors. For example, teacher effects on students’ *Self-Efficacy in Math* and teacher effects on the other two non-tested outcomes fall between 0.44 and 0.65. However, teachers do not appear to be equally effective at improving all three attitudes and behaviors. The correlation between teacher effects on students’ *Happiness in Class* and teacher effects on students’ *Behavior in Class* is weak and nonsignificant. Correlations between teacher effects on students’ attitudes and behaviors versus effects on students’ math achievement are similarly weak; most are not statistically significant. One exception is the relationship between teacher effects on students’ math performance and teacher effects on students’ *Happiness in Class*, which is negative and statistically significantly correlated when using the shrunken estimates (*r* = −0.38). This suggests teachers who are skilled at improving students’ math achievement may do so in ways that make students less happy or less engaged in class.

Overall, these findings provide strong evidence that teachers impact several student attitudes and behaviors in addition to their academic performance. Weak, nonsignificant correlations between many of these teacher effect estimates indicate these measures identify unique skills that teachers bring to and engage in the classroom.

### Sensitivity of Teacher Effects Across Model Specifications

In tables 6 and 7, I present results describing the relationship between teacher effects on students’ attitudes and behaviors across model specifications. As before, panel A shows correlations for unshrunken estimates, and panel B shows correlations for shrunken estimates. Because patterns of results are quite similar for the unshrunken and shrunken estimates, I focus my discussion on the latter for simplicity. This analysis includes the balanced sample of teachers and students with all possible control variables from the larger observational dataset. In Appendix table A.4, I present the magnitude of the teacher-level variation on all student outcomes using this sample, and find that results are similar to those presented in table 3 using the experimental sample.

. | $\rho Model1,Model2$ . | $\rho Model1,Model2$ . | $\rho Model2,Model3$ . |
---|---|---|---|

Panel A: Unshrunken Estimates . | |||

Teacher effects on Behavior in Class | 0.89^{***} | 0.89^{***} | 1.00^{***} |

Teacher effects on Self-Efficacy in Math | 0.88^{***} | 0.91^{***} | 0.98^{***} |

Teacher effects on Happiness in Class | 0.96^{***} | 0.97^{***} | 0.99^{***} |

Panel B: Shrunken Estimates | |||

Teacher effects on Behavior in Class | 0.90^{***} | 0.91^{***} | 1.00^{***} |

Teacher effects on Self-Efficacy in Math | 0.86^{***} | 0.90^{***} | 0.97^{***} |

Teacher effects on Happiness in Class | 0.96^{***} | 0.96^{***} | 0.99^{***} |

. | $\rho Model1,Model2$ . | $\rho Model1,Model2$ . | $\rho Model2,Model3$ . |
---|---|---|---|

Panel A: Unshrunken Estimates . | |||

Teacher effects on Behavior in Class | 0.89^{***} | 0.89^{***} | 1.00^{***} |

Teacher effects on Self-Efficacy in Math | 0.88^{***} | 0.91^{***} | 0.98^{***} |

Teacher effects on Happiness in Class | 0.96^{***} | 0.97^{***} | 0.99^{***} |

Panel B: Shrunken Estimates | |||

Teacher effects on Behavior in Class | 0.90^{***} | 0.91^{***} | 1.00^{***} |

Teacher effects on Self-Efficacy in Math | 0.86^{***} | 0.90^{***} | 0.97^{***} |

Teacher effects on Happiness in Class | 0.96^{***} | 0.96^{***} | 0.99^{***} |

*Notes:* Model 1 calculates teacher effectiveness ratings that control for students' prior achievement in math and reading. Model 2 controls for a prior measure of students’ attitude or behavior. Model 3 controls for prior scores on both prior achievement and prior attitude or behavior. Samples include 51 teachers.

^{***}*p* < 0.001.

. | $\rho Model1,Model4$ . | $\rho Model1,Model5$ . | $\rho Model1,Model6$ . | $\rho Model1,Model7$ . |
---|---|---|---|---|

Panel A: Unshrunken Estimates . | ||||

Teacher effects on state math test | 0.98^{***} | 0.72^{***} | 0.64^{***} | 0.49^{***} |

Teacher effects on Behavior in Class | 0.95^{***} | 0.74^{***} | 0.64^{***} | 0.38^{***} |

Teacher effects on Self-Efficacy in Math | 0.99^{***} | 0.84^{***} | 0.78^{***} | 0.46^{***} |

Teacher effects on Happiness in Class | 0.97^{***} | 0.85^{***} | 0.52^{***} | 0.49^{***} |

Panel B: Shrunken EB Estimates | ||||

Teacher effects on state math test | 0.99^{***} | 0.76^{***} | 0.54^{***} | 0.53^{***} |

Teacher effects on Behavior in Class | 0.98^{***} | 0.69^{***} | 0.63^{***} | 0.41^{***} |

Teacher effects on Self-Efficacy in Math | 0.99^{***} | — | — | — |

Teacher effects on Happiness in Class | 0.99^{***} | 0.90^{***} | 0.71^{***} | 0.66^{***} |

. | $\rho Model1,Model4$ . | $\rho Model1,Model5$ . | $\rho Model1,Model6$ . | $\rho Model1,Model7$ . |
---|---|---|---|---|

Panel A: Unshrunken Estimates . | ||||

Teacher effects on state math test | 0.98^{***} | 0.72^{***} | 0.64^{***} | 0.49^{***} |

Teacher effects on Behavior in Class | 0.95^{***} | 0.74^{***} | 0.64^{***} | 0.38^{***} |

Teacher effects on Self-Efficacy in Math | 0.99^{***} | 0.84^{***} | 0.78^{***} | 0.46^{***} |

Teacher effects on Happiness in Class | 0.97^{***} | 0.85^{***} | 0.52^{***} | 0.49^{***} |

Panel B: Shrunken EB Estimates | ||||

Teacher effects on state math test | 0.99^{***} | 0.76^{***} | 0.54^{***} | 0.53^{***} |

Teacher effects on Behavior in Class | 0.98^{***} | 0.69^{***} | 0.63^{***} | 0.41^{***} |

Teacher effects on Self-Efficacy in Math | 0.99^{***} | — | — | — |

Teacher effects on Happiness in Class | 0.99^{***} | 0.90^{***} | 0.71^{***} | 0.66^{***} |

*Notes:* Baseline model to which others are compared (model 1) calculates teacher effectiveness ratings that only control for students’ prior achievement in math and reading. Model 4 adds student demographic characteristics, including gender, race, free or reduced-price lunch eligibility, special education status, and limited English proficiency status; Model 5 adds classroom characteristics. Model 6 adds school characteristics. Model 7 replaces school characteristics with school fixed effects. Empty cells indicate that variation in teacher effects from one of the models is close to 0 SD (see Appendix table A.4). Samples include 51 teachers.

^{***}*p* < 0.001.

In the first of these tables (table 6), I examine the correlations between teacher effects on students’ attitudes and behaviors that are estimated controlling for prior achievement (model 1), a prior measure of the survey outcome (model 2), or both (model 3). Because this analysis examines the sensitivity of teacher effects to inclusion or exclusion of lagged measures of the outcome variable, I focus only on teacher effects on the three measures of students’ attitudes and behaviors. For teacher effects on students’ math performance used elsewhere in the paper, all models control for lagged achievement.

Here, I find correlations of teacher effects across model specifications above 0.86. As expected, the smallest of these correlations describe the relationship between teacher effects that control either for prior achievement (model 1) or for students’ prior survey responses (model 2). However, these correlations still are quite strong: 0.90 for teacher effects on *Behavior in Class*, 0.86 for *Self-Efficacy in Math*, and 0.96 for *Happiness in Class*. Correlations between teacher effects from models that have overlapping sets of controls (i.e., between models 1 and 3 or between models 2 and 3) are stronger, between 0.90 and 0.99. This suggests teacher effects on these attitudes and behaviors are not particularly sensitive to inclusion of prior achievement or prior survey responses. In light of these findings, I exclude prior measures of students’ attitudes and behaviors from most subsequent analyses, allowing me to retain the largest possible sample of teachers and students.

Next, I examine the sensitivity of teacher effects from this baseline model (model 1) to models that control for additional student, class, or school characteristics (see table 7). In the table, empty cells indicate instances where the teacher-level variation is close to 0 SD (i.e., for shrunken teacher effects on students’ *Self-Efficacy in Math* generated from models 5 through 7; see Appendix table A.4). I find that teacher effects on students’ math performance and on the three measures of students’ attitudes and behaviors are not particularly sensitive to student demographic characteristics but are sensitive to additional control variables. Correlations between teacher effect estimates from model 1 (which controls for prior test scores) and model 4 (which builds on model 1 by adding student demographic characteristics) all are greater than or equal to 0.95. For teacher effects on students’ math performance, *Behavior in Class*, and *Self-Efficacy in Math,* correlations between estimates from model 1 and from model 5 (which builds on previous models by adding classroom characteristics) are substantively smaller, at 0.76, 0.69, and 0.82, respectively. For teacher effects on students’ *Happiness in Class*, the correlation stays above 0.90.

Adding school characteristics to teacher effect specifications appears to have the largest impact on teacher rankings. Correlations between estimates from model 1 and from model 6 (which builds on previous models by adding observable school characteristics) range from 0.54 (for teacher effects on students’ math performance) to 0.71 (for teacher effects on students’ *Happiness in Class*). Correlations between estimates from model 1 and from model 7 (which replaces observable school characteristics with school fixed effects) range from 0.41 (for teacher effects on students’ *Behavior in Class*) to 0.66 (for teacher effects on students’ *Happiness in Class*). Correlations between estimates from models 6 and 7 (not shown in table 7) are 0.94, 0.70, 0.71, and 0.92 for teacher effects on students’ math performance, *Behavior in Class, Self-Efficacy in Math,* and *Happiness in Class*, respectively. Reference bias is one possible explanation for lower correlations in models that do and do not control for school fixed effects. At the same time, these estimates are well within the range reported in studies looking at the sensitivity of teacher effects on test scores across models that control for school characteristics or school fixed effects, between roughly 0.5 and 0.9 (Aaronson, Barrow, and Sander 2007; Hill, Kapitula, and Umland 2011; Goldhaber and Theobald 2012).

### Predictive Validity of Nonexperimental Teacher Effects

In table 8, I report estimates describing the relationship between nonexperimental teacher effects on student outcomes and these same measures following random assignment. Cells contain estimates from separate regression models where the dependent variable is the student attitude or behavior listed in each column. The independent variable of interest is the nonexperimental teacher effect on this same outcome estimated in years prior to random assignment. Nonexperimental teacher effects are modeled from five separate equations discussed above, each with different sets of covariates. I exclude teacher effects calculated from models 2 and 3 (described in table 6), both of which controlled for prior measures of students’ attitudes and behaviors that were not available for many teachers’ students in the experimental portion of the study. However, at the end of this section I describe results from additional analyses that estimate the relationship between nonexperimental teacher effects that control for imputed lagged survey measures; results are consistent with the main results. Stars indicate whether point estimates are statistically significantly different from 0 SD, and *p*-values testing the null hypothesis that effect sizes are equal to 1 SD are presented next to each estimate. The sample sizes for *Happiness in Class* is reduced by one teacher who did not have nonexperimental teacher effects on this outcome.

. | State Math Test . | Behavior in Class . | Self-Efficacy in Math . | Happiness in Class . | ||||
---|---|---|---|---|---|---|---|---|

. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. |

Panel A: Unshrunken Estimates . | ||||||||

Teacher effects calculated from model 1 | 0.846^{***} | 0.046 | 0.685^{***} | 0.064 | 0.418^{∼} | 0.010 | 0.350^{*} | 0.000 |

(0.075) | (0.165) | (0.216) | (0.142) | |||||

Teacher effects calculated from model 4 | 0.869^{***} | 0.115 | 0.706^{***} | 0.054 | 0.414^{∼} | 0.011 | 0.361^{*} | 0.000 |

(0.081) | (0.148) | (0.219) | (0.140) | |||||

Teacher effects calculated from model 5 | 0.914^{***} | 0.366 | 0.719^{***} | 0.058 | 0.447^{∼} | 0.029 | 0.349^{*} | 0.000 |

(0.094) | (0.144) | (0.244) | (0.133) | |||||

Teacher effects calculated from model 6 | 0.915^{***} | 0.361 | 0.742^{***} | 0.077 | 0.461^{∼} | 0.035 | 0.399^{**} | 0.000 |

(0.092) | (0.142) | (0.247) | (0.136) | |||||

Teacher effects calculated from model 7 | 0.903^{***} | 0.314 | 0.768^{***} | 0.118 | 0.448^{∼} | 0.023 | 0.399^{**} | 0.000 |

(0.095) | (0.145) | (0.234) | (0.134) | |||||

Panel B: Shrunken Estimates | ||||||||

Teacher effects calculated from model 1 | 0.960^{***} | 0.609 | 1.003^{***} | 0.992 | 0.514 | 0.195 | 0.427^{*} | 0.003 |

(0.078) | (0.266) | (0.369) | (0.177) | |||||

Teacher effects calculated from model 4 | 0.995^{***} | 0.953 | 1.090^{***} | 0.738 | 0.507 | 0.192 | 0.438^{*} | 0.003 |

(0.084) | (0.268) | (0.372) | (0.175) | |||||

Teacher effects calculated from model 5 | 1.055^{***} | 0.585 | 1.240^{***} | 0.436 | — | 0.416^{*} | 0.001 | |

(0.100) | (0.305) | (0.167) | ||||||

Teacher effects calculated from model 6 | 1.079^{***} | 0.435 | 1.472^{***} | 0.207 | — | 0.487^{**} | 0.005 | |

(0.101) | (0.368) | (0.174) | ||||||

Teacher effects calculated from model 7 | 1.084^{***} | 0.419 | 1.789^{***} | 0.092 | — | 0.522^{**} | 0.008 | |

(0.102) | (0.458) | (0.172) | ||||||

Teachers | 41 | 41 | 41 | 40 | ||||

Students | 531 | 531 | 531 | 509 |

. | State Math Test . | Behavior in Class . | Self-Efficacy in Math . | Happiness in Class . | ||||
---|---|---|---|---|---|---|---|---|

. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. |

Panel A: Unshrunken Estimates . | ||||||||

Teacher effects calculated from model 1 | 0.846^{***} | 0.046 | 0.685^{***} | 0.064 | 0.418^{∼} | 0.010 | 0.350^{*} | 0.000 |

(0.075) | (0.165) | (0.216) | (0.142) | |||||

Teacher effects calculated from model 4 | 0.869^{***} | 0.115 | 0.706^{***} | 0.054 | 0.414^{∼} | 0.011 | 0.361^{*} | 0.000 |

(0.081) | (0.148) | (0.219) | (0.140) | |||||

Teacher effects calculated from model 5 | 0.914^{***} | 0.366 | 0.719^{***} | 0.058 | 0.447^{∼} | 0.029 | 0.349^{*} | 0.000 |

(0.094) | (0.144) | (0.244) | (0.133) | |||||

Teacher effects calculated from model 6 | 0.915^{***} | 0.361 | 0.742^{***} | 0.077 | 0.461^{∼} | 0.035 | 0.399^{**} | 0.000 |

(0.092) | (0.142) | (0.247) | (0.136) | |||||

Teacher effects calculated from model 7 | 0.903^{***} | 0.314 | 0.768^{***} | 0.118 | 0.448^{∼} | 0.023 | 0.399^{**} | 0.000 |

(0.095) | (0.145) | (0.234) | (0.134) | |||||

Panel B: Shrunken Estimates | ||||||||

Teacher effects calculated from model 1 | 0.960^{***} | 0.609 | 1.003^{***} | 0.992 | 0.514 | 0.195 | 0.427^{*} | 0.003 |

(0.078) | (0.266) | (0.369) | (0.177) | |||||

Teacher effects calculated from model 4 | 0.995^{***} | 0.953 | 1.090^{***} | 0.738 | 0.507 | 0.192 | 0.438^{*} | 0.003 |

(0.084) | (0.268) | (0.372) | (0.175) | |||||

Teacher effects calculated from model 5 | 1.055^{***} | 0.585 | 1.240^{***} | 0.436 | — | 0.416^{*} | 0.001 | |

(0.100) | (0.305) | (0.167) | ||||||

Teacher effects calculated from model 6 | 1.079^{***} | 0.435 | 1.472^{***} | 0.207 | — | 0.487^{**} | 0.005 | |

(0.101) | (0.368) | (0.174) | ||||||

Teacher effects calculated from model 7 | 1.084^{***} | 0.419 | 1.789^{***} | 0.092 | — | 0.522^{**} | 0.008 | |

(0.102) | (0.458) | (0.172) | ||||||

Teachers | 41 | 41 | 41 | 40 | ||||

Students | 531 | 531 | 531 | 509 |

*Notes:* Cells include estimates from separate regression models that control for students’ prior achievement in math and reading, student demographic characteristics, classroom characteristics from randomly assigned rosters, and fixed effects for randomization block. Robust standard errors clustered at the class level in parentheses. Model 1 calculates teacher effectiveness ratings that only control for students’ prior achievement in math and reading; Model 4 adds student demographic characteristics; Model 5 adds classroom characteristics; Model 6 adds school characteristics; Model 7 replaces school characteristics with school fixed effects. Empty cells indicate that variation in nonexperimental teacher effects is close to 0 SD (see Appendix table A.4).

^{∼}*p* <0.10; ^{*}*p* < 0.05; ^{**}*p* < 0.01; ^{***}*p* < 0.001.

Validity evidence for teacher effects on students’ math performance are consistent with other experimental studies (Kane and Staiger 2008; Kane et al. 2013), where predicted differences in teacher effectiveness in observational data are equal to or come close to actual differences following random assignment of teachers to classes. The nonexperimental teacher effect estimate that comes closest to a 1:1 relationship is the shrunken estimate that controls for students’ prior achievement and other demographic characteristics (0.995 SD). Despite a relatively small sample of teachers, the standard error for this estimate (0.084) is substantively smaller than those in other experimental studies—including the meta-analysis conducted by Bacher-Hicks et al. (2017)—and allows me to rule out relatively large degrees of bias in teacher effects calculated from this model. A likely explanation for greater precision in this study relative to others is the fact that other studies generate estimates through instrumental variables estimation to calculate treatment on the treated. Instead, I use OLS regression and account for noncompliance by narrowing in on randomization blocks in which very few, if any, students moved out of their randomly assigned teachers’ classroom. Nonexperimental teacher effects calculated without shrinkage are related less strongly to current student outcomes, though differences in estimates and associated standard errors between panel A and panel B are not large. All corresponding estimates (e.g., model 1 from panel A versus panel B) have overlapping 95 percent confidence intervals.

Results examining forecast bias in teacher effects on students’ *Behavior in Class* are not substantively different from what I would expect based on the math test-score outcome. I find that teacher effects with the best predictive validity are the shrunken estimates from model 1, which calculates nonexperimental teacher effects only controlling for students’ prior achievement.^{12} Here, I find an estimate of 1.00 SD that matches the hypothesis described by Kane and Staiger (2008), where predicted differences across classroom should equal observed differences. However, the standard error around this estimate is substantively larger (0.25) than the standard error for test-score estimates.

Comparison of estimates between panel A and panel B provides some insight here and, in particular, the tradeoff between accuracy and precision. In panel B, estimates relating nonexperimental, shrunken teacher effect estimates on students’ *Behavior in Class* to current student outcomes are notably larger and closer to 1 SD than estimates in panel A relating unshrunken estimates to current student outcomes. This makes sense, as shrunken estimates are adjusted for the amount of measurement error the unshrunken estimates contain. Measurement error will attenuate the relationship between two teacher effect estimates, even if the true relationship is equal to 1 SD. Indeed, earlier in the paper, I showed that teacher effects on students’ *Behavior in Class* generally underwent more shrinkage than teacher effects on students’ math test scores (see table 3). At the same time, relationships between shrunken teacher effect estimates and current student outcomes in panel B are measured with considerably less precision than relationships drawing on unshrunken teacher effect estimates in panel A. Standard errors in panel B are roughly two to three times as large as those in panel A. This also makes sense, as shrunken estimates provide a lower bound on the variation of true teacher effects, particularly when measurement error is large (Jacob and Lefgren 2005); decreased variation in the independent variable decreases statistical power. Considering results from panels A and B jointly provides evidence that nonexperimental methods for estimating teacher effects on students’ *Behavior in Class* account for a large degree of bias due to nonrandom sorting and factors beyond teachers’ control.

For both *Self-Efficacy in Math* and *Happiness in Class*, nonexperimental teacher effect estimates have moderate predictive validity. Generally, I can distinguish estimates from 0 SD, indicating they contain some information content on teachers. The exception is shrunken estimates for *Self-Efficacy in Math*. Although estimates are similar in magnitude to the unshrunken estimates in panel A, between 0.42 SD and 0.58 SD standard errors are large and 95 percent confidence intervals cross 0 SD. I also can distinguish many estimates from 1 SD. This indicates that nonexperimental teacher effects on students’ *Self-Efficacy in Math* and *Happiness in Class* contain potentially large and important degrees of bias. For both measures of teacher effectiveness, point estimates around 0.5 SD suggest that they contain roughly 50 percent bias.

One concern with these results is that nonexperimental teacher effects do not control for prior survey responses, and such a model might reduce bias. Earlier in the paper, I show that in the observational portion of the study, teacher effects that control for some combination of prior achievement and/or prior survey responses do not return markedly different teacher rankings. However, in some instances correlations are lower than 0.90 (see table 6), leaving open the possibility that controlling for lagged survey responses may boost performance. To address this concern in the experimental data, I conduct a robustness check that calculates nonexperimental teacher effects after imputing students’ lagged survey responses. To impute, I fit a series of regression models that predict students’ lagged survey response with all other available data (i.e., prior test scores in math and reading, and the student demographic characteristics listed in table 2) for the sample of students with these data used elsewhere in this paper.^{13} Then, I use this model to infer predicted values for all students without lagged survey responses. Finally, I calculate nonexperimental teacher effects on students’ attitudes and behaviors controlling for these lagged measures, an indicator for whether or not the lagged survey measure was imputed, and in some instances, students’ prior test scores. Model 2 calculates nonexperimental teacher effects controlling for students’ prior survey response, while model 3 calculates nonexperimental teacher effects controlling for prior achievement and prior survey response. I present results that relate these teacher effects to student outcomes following random assignment in Appendix table A.5. Patterns of results are very similar to those presented in table 8, suggesting that controlling for lagged measures of the outcome variable when calculating nonexperimental teacher effects on students’ attitudes and behaviors does not appear to change inferences regarding bias in these measures. These results also are similar to those from the quasi-experimental validation study by Backes and Hansen (2018), where the authors controlled for prior measures of their non-tested outcomes in all models and still found large degrees of bias in some instances.

## 7. Discussion and Conclusion

Where does this leave policy, practice, and research? Should these measures be used in policy settings, despite concerns about bias? This is not an easy question to answer. For some readers, the relationships presented in this paper could point to considerable policy usefulness for the nonexperimental estimates. If one were to rank teachers using experimental and nonexperimental estimates, results would be similar. Thus, ignoring reference bias problems and possible gaming, a teacher de-selection policy using biased measures would still improve outcomes on average.

Another possible reason to incorporate measures of students’ attitudes and behaviors and teachers’ ability to improve them into selection and accountability policy, in spite of bias, would be to create clear incentives for improving these skills in school. Many, including myself, see students’ social and emotional development as a central goal of teachers’ and schools’ work (e.g., Pianta and Hamre 2009; Durlak et al. 2011; Farrington et al. 2012). Yet, accountability systems that focus predominantly or exclusively on student achievement send a message that the skills captured on these tests are the ones that policy makers want students to have when they leave school. Broadening what it means to be a successful student and “making the development of the whole child central to the mission of education” (Garcia 2014, p. 4) clearly is good policy.

At the same time, lessons learned from new teacher evaluation systems that incorporate teacher effects on students’ test scores highlight several reasons why making high-stakes policy decisions based on teacher effects on students’ attitudes and behaviors may not be appropriate or advantageous. Despite convincing evidence against bias in teacher effects on students’ academic performance, teachers still are skeptical about their use and the fairness of these measures (Jiang, Sporte, and Luppescu 2015). One reason for this skepticism discussed in the academic literature is that, even if teacher effects are unbiased, they often are quite noisy measures of teachers’ effectiveness (Ballou and Springer 2015). Large confidence intervals around individual teachers’ scores—due to the number of students attached to that teacher and error in the student-level assessment itself—mean that a teacher's underlying ability often is statistically indistinguishable from other teachers. This likely would be an even greater issue for teacher effects on students’ attitudes and behaviors given well-documented concerns about error in student-, teacher-, and parent-reports of these measures (Duckworth and Yeager 2015). Even if systems were to incorporate classical measurement error into teachers’ effectiveness ratings, which most do not, there still would be lingering concerns about other sources of error. In particular, there are bound to be concerns about cheating (Campbell 1979; Koretz 2008). In this study, student surveys were administered under low-stakes conditions where student responses were not visible to the teacher or other students in the classroom. It is possible that estimates of bias might differ—likely to increase—under high-stakes settings where survey responses could be coached or influenced by other pressures.

Despite concern about using teacher effects on students’ attitudes and behaviors in high-stakes policy settings, I believe there are other uses of these measures that fall within and would enhance existing school practices. In particular, measures of teachers’ effectiveness at improving students’ attitudes and behaviors could be used to identify areas for professional growth and connect teachers with targeted professional development. Bringing costly but effective development programs, such as teacher coaching (Kraft, Blazar, and Hogan 2018), to scale requires at least two key pieces of information that measures such as those used in this study could provide. First, it would be useful to know which teachers require immediate support in order to allocate professional development dollars to these teachers, as opposed to investing in lower-cost but less-effective programs that reach all teachers. Second, the individualized nature of coaching and related development programs require that school leaders know teachers’ individual strengths and weaknesses in order to facilitate appropriate teacher-coach or teacher-team matches, where members have complementary skill sets (Papay et al. 2016). Observation rubrics provide one source of data for this purpose, yet have logistical constraints, including needing school leaders who have the time and knowledge to assess multiple teachers on multiple teaching skills (Hill and Grossman 2013). In light of moderate to strong relationships between teachers’ observed classroom behaviors captured on established observation rubrics, and teacher effects on several student attitudes and behaviors (Blazar and Kraft 2017), it is possible that the latter could be used as a lower-cost proxy for the former. In these instances, biased measures are less likely to be a concern than in settings where teachers’ jobs are on the line.

Finally, supporting teachers in the work of developing students’ attitudes and behaviors will require investments in research in addition to changes in policy and practice. Based on experimental studies of teacher effects on student achievement, newer research is starting to examine related questions (e.g., how instructional supports impact students’ achievement; Kane et al. 2016) using observational and value-added approaches that generally are less expensive and considerably more tractable than randomized control trials. Making this important decision in the context of research on students’ attitudes and behaviors will require close consideration of the tradeoffs between two key issues: bias and availability of data. The analyses presented here suggest that value-added approaches likely will reduce some but not all of the sorting bias that could influence estimates of the impact of different inputs on measures of students’ behavior, self-efficacy, and happiness. Even if some degree of bias remains, this approach likely would improve upon much of the existing body of research to date that lacks convincing evidence about what works in education (Murnane and Willett 2011; Kane 2015). At the same time, such studies would not have the benefit of easily accessible administrative data that have made this type of work possible when examining gains in student achievement outcomes. Building up administrative datasets that include rich measures of students’ attitudes and behaviors in addition to their academic performance is, in my opinion, a worthy goal (see West 2016 for how this is starting to happen in some education agencies, including the CORE districts in California). Until that happens, though, researchers will need to continue to collect these measures themselves. In turn, we likely will want to conduct more and learn as much as possible from random assignment studies.

## Acknowledgments

I would like to thank Martin West, Thomas Kane, Heather Hill, and anonymous reviewers for their feedback on earlier drafts. The data collection effort described here was supported by the Institute of Education Sciences, U.S. Department of Education, through grant R305C090023 to the President and Fellows of Harvard College to support the National Center for Teacher Effectiveness. The opinions expressed are those of the author and do not represent views of the Institute or the U.S. Department of Education. Research support came from the Mathematica Policy Research summer fellowship and the Smith Richardson Foundation.

## REFERENCES

## Notes

See Bacher-Hicks et al. (2017) for an analysis of persistent sorting in the classroom data used in this study.

In a fourth experimental study, Glazerman and Protik (2015) exploited random assignment of teachers across schools as part of a merit pay program. There, findings were more mixed. In the elementary school sample, the authors estimated a standardized effect size of roughly 1 SD relating nonexperimental value-added scores (stacking across math and reading) to student test scores following random assignment. However, in their smaller sample, the standard error was large (0.34), meaning they could not rule out potentially large degrees of bias. Further, in the middle school sample, they found no statistically significant relationship between nonexperimental and experimental teacher effect estimates.

Two additional studies have examined teacher effects on students’ attitudes and behaviors using the random assignment portion of the Measures of Effective Teaching (MET) project. Kraft (forthcoming) found sizeable teacher effects on students’ grit, growth mindset, and effort in class (0.10 to 0.17 SD). Correlations between teacher effects on students’ academic performance versus effects on other outcomes were no higher than 0.22. Kane et al. (2013) found that a composite measure of teacher effectiveness based on observational data predicted student effort following random assignment. However, measures of students’ attitudes and behaviors were collected in only one year. Therefore, it was not possible to relate teacher effects calculated under nonexperimental conditions to teacher effects on this same outcome calculated under experimental ones.

For all three outcomes, composite scores that average across raw responses are correlated at 0.99 and above with scales that incorporate weights from the factor analysis.

This sample size is driven by teachers whose students had current- and prior-year survey responses for *Happiness in Class*, which was only available in two of the three years of the study. Additional teachers and students had current- and prior-year data for *Behavior in Class* (*N* = 111) and *Self-Efficacy in Math* (*N* = 108), both of which were available in all three years of the study. For consistency, I limit this sample to teachers and students who had current- and prior-year scores for all three survey measures. I did not place any restriction on the number of students each teacher needed to have in order to be included in this sample. Although others advocate excluding teachers with fewer than five students per class (Kane and Staiger 2008), for example, this would have further reduced my sample. Instead, I rely on the fact that shrinkage estimators shrink estimates more for teachers with few students than for teachers with larger classes.

Background information on teachers was captured on a questionnaire administered in the fall of each year. Survey items included years teaching math, route to certification, amount of undergraduate or graduate coursework in math and math courses for teaching (1 = No classes, 2 = One or two classes, 3 = Three to five classes, 4 = Six or more classes). For simplicity, I averaged these last two items to form one construct capturing teachers’ mathematics coursework. Further, the survey included a test of teachers’ mathematical content knowledge, with items from both the Mathematical Knowledge for Teaching assessment and the Massachusetts Test for Educator Licensure. Teacher scores were generated by IRTPro software and standardized in these models, with a reliability of 0.92. For more information about these constructs, see Hill, Blazar, and Lynch 2015.

Two other teachers from the same randomization block also agreed to participate. However, the principal decided that it was not possible to randomly assign rosters to these teachers. Thus, I exclude them from all analyses.

One concern with dropping teachers in this way is that they may differ from other teachers on post-randomization outcomes, which could bias results. Comparing attriters for whom I had post-randomization data (*N* = 21, which excludes the four teachers who either left teaching, left the district, moved to third grade and therefore out of my dataset, or were on maternity leave) to the remaining teachers (*N* = 54) on their observed effectiveness at raising students’ math achievement in the 2012–13 school year, I found no difference (*p* = 0.899). Further, to ensure strong external validity, I compared attriters to the experimental sample on each of the teacher characteristics listed in table 1 and found no difference on any.

Thirty-eight students were hand-placed in these teachers’ classrooms after the random assignment process. As these students were not part of the experiment, they were excluded from all analyses.

Twenty-six students were missing baseline data on at least one characteristic. In order to retain all students, I imputed missing data to the mean of the students’ randomization block. I take the same approach to missing data in all subsequent analyses. This includes the nineteen students who were part of my main analytic sample but happened to be absent on the day that project managers administered the student survey and, thus, were missing outcome data. This approach to imputation seems reasonable given that there was no reason to believe that students were absent on purpose to avoid taking the survey.

An alternative fixed-effects specification is preferred by some because it does not assume that teacher assignment is uncorrelated with factors that predict student outcomes (Guarino et al. 2015). However, in these data, this approach returned similar estimates in models where it was feasible to include teacher fixed effects in addition to the other set of control variables, with correlations of 0.99 or above (see Blazar and Kraft 2017 for more details).

Results in panel B suggest that adding class- and school-level controls to the model calculating nonexperimental teacher effects on students’ *Behavior in Class* may in fact add bias. Point estimates describing the relationship between current student outcomes and nonexperimental teacher effects calculated from models 3 through 5 all are greater than 1.2 SD. These patterns are somewhat consistent with findings from the MET project, in which researchers suggested that some models “over control,” resulting in removal of peer effects that actually predict important differences in teacher performance (Kane et al. 2013). This explanation makes sense here as well, where teachers’ ability to improve individual students’ behavior likely is closely related to the control they have over other peer-to-peer relationships. At the same time, I do not want to place too much emphasis on these differences across models, given that standard errors are large and, thus, point estimates have overlapping 95 percent confidence intervals.

Students’ prior test scores are statistically significant predictors of all three measures of students’ attitudes and behavior, as are several demographic characteristics. Together, prior achievement and demographic characteristics explain a sizeable amount of the variation in prior survey responses: 22 percent for *Behavior in Class*, 10 percent for *Self-Efficacy in Math*, and 7 percent for *Happiness in Class*. Further, imputed measures of students’ prior survey responses are moderately to strongly related to current survey responses, with standardized regression coefficients between 0.43 SD (*Happiness in Class*) and 0.63 SD (*Behavior in Class*).

## Appendix A: Additional Data

. | Univariate Statistics . | Pairwise Correlations . | ||||
---|---|---|---|---|---|---|

. | Mean . | SD . | Cronbach's Alpha . | Behavior in Class . | Self-Efficacy in Math . | Happiness in Class . |

Behavior in Class | 4.10 | 0.93 | 0.74 | 1.00 | ||

My behavior in this class is good. | 4.23 | 0.89 | ||||

My behavior in this class sometimes annoys the teacher. | 3.80 | 1.35 | ||||

My behavior is a problem for the teacher in this class. | 4.27 | 1.13 | ||||

Self-Efficacy in Math | 4.17 | 0.58 | 0.76 | 0.35^{***} | 1.00 | |

I have pushed myself hard to completely understand math in this class. | 4.23 | 0.97 | ||||

If I need help with math, I make sure that someone gives me the help I need. | 4.12 | 0.97 | ||||

If a math problem is hard to solve, I often give up before I solve it. | 4.26 | 1.15 | ||||

Doing homework problems helps me get better at doing math. | 3.86 | 1.17 | ||||

In this class, math is too hard. | 4.05 | 1.10 | ||||

Even when math is hard, I know I can learn it. | 4.49 | 0.85 | ||||

I can do almost all the math in this class if I don't give up. | 4.35 | 0.95 | ||||

I'm certain I can master the math skills taught in this class. | 4.24 | 0.90 | ||||

When doing work for this math class, focus on learning not time work takes. | 4.11 | 0.99 | ||||

I have been able to figure out the most difficult work in this math class. | 3.95 | 1.09 | ||||

Happiness in Class | 4.10 | 0.85 | 0.82 | 0.27^{***} | 0.62^{***} | 1.00 |

This math class is a happy place for me to be. | 3.98 | 1.13 | ||||

Being in this math class makes me feel sad or angry. | 4.38 | 1.11 | ||||

The things we have done in math this year are interesting. | 4.04 | 0.99 | ||||

Because of this teacher, I am learning to love math. | 4.02 | 1.19 | ||||

I enjoy math class this year. | 4.12 | 1.13 |

. | Univariate Statistics . | Pairwise Correlations . | ||||
---|---|---|---|---|---|---|

. | Mean . | SD . | Cronbach's Alpha . | Behavior in Class . | Self-Efficacy in Math . | Happiness in Class . |

Behavior in Class | 4.10 | 0.93 | 0.74 | 1.00 | ||

My behavior in this class is good. | 4.23 | 0.89 | ||||

My behavior in this class sometimes annoys the teacher. | 3.80 | 1.35 | ||||

My behavior is a problem for the teacher in this class. | 4.27 | 1.13 | ||||

Self-Efficacy in Math | 4.17 | 0.58 | 0.76 | 0.35^{***} | 1.00 | |

I have pushed myself hard to completely understand math in this class. | 4.23 | 0.97 | ||||

If I need help with math, I make sure that someone gives me the help I need. | 4.12 | 0.97 | ||||

If a math problem is hard to solve, I often give up before I solve it. | 4.26 | 1.15 | ||||

Doing homework problems helps me get better at doing math. | 3.86 | 1.17 | ||||

In this class, math is too hard. | 4.05 | 1.10 | ||||

Even when math is hard, I know I can learn it. | 4.49 | 0.85 | ||||

I can do almost all the math in this class if I don't give up. | 4.35 | 0.95 | ||||

I'm certain I can master the math skills taught in this class. | 4.24 | 0.90 | ||||

When doing work for this math class, focus on learning not time work takes. | 4.11 | 0.99 | ||||

I have been able to figure out the most difficult work in this math class. | 3.95 | 1.09 | ||||

Happiness in Class | 4.10 | 0.85 | 0.82 | 0.27^{***} | 0.62^{***} | 1.00 |

This math class is a happy place for me to be. | 3.98 | 1.13 | ||||

Being in this math class makes me feel sad or angry. | 4.38 | 1.11 | ||||

The things we have done in math this year are interesting. | 4.04 | 0.99 | ||||

Because of this teacher, I am learning to love math. | 4.02 | 1.19 | ||||

I enjoy math class this year. | 4.12 | 1.13 |

*Notes:* Statistics are generated from all available data. All survey items are on a scale from 1 to 5. Statistics drawn from all available data.

^{***}*p* < 0.001.

. | Number of Students . | Percent of Total . |
---|---|---|

Remained with randomly assigned teacher | 677 | 72 |

Switched teacher within school | 168 | 18 |

Left school | 40 | 4 |

Left district | 49 | 5 |

Unknown | 9 | 1 |

Total | 943 | 100 |

. | Number of Students . | Percent of Total . |
---|---|---|

Remained with randomly assigned teacher | 677 | 72 |

Switched teacher within school | 168 | 18 |

Left school | 40 | 4 |

Left district | 49 | 5 |

Unknown | 9 | 1 |

Total | 943 | 100 |

. | Noncompliers . | Compliers . | p-value on Difference
. |
---|---|---|---|

Student Characteristics | |||

Male | 0.38 | 0.49 | 0.044 |

African American | 0.38 | 0.33 | 0.374 |

Asian | 0.12 | 0.15 | 0.435 |

Hispanic | 0.15 | 0.21 | 0.128 |

White | 0.31 | 0.27 | 0.403 |

FRPL | 0.64 | 0.66 | 0.572 |

SPED | 0.06 | 0.05 | 0.875 |

LEP | 0.11 | 0.21 | 0.016 |

Prior achievement on state math test | 0.30 | 0.26 | 0.689 |

Prior achievement on state reading test | 0.28 | 0.30 | 0.782 |

p-value on joint test | 0.146 | ||

Teacher characteristics | |||

Prior teacher effects on state math scores | −0.01 | −0.01 | 0.828 |

Students | 67 | 531 |

. | Noncompliers . | Compliers . | p-value on Difference
. |
---|---|---|---|

Student Characteristics | |||

Male | 0.38 | 0.49 | 0.044 |

African American | 0.38 | 0.33 | 0.374 |

Asian | 0.12 | 0.15 | 0.435 |

Hispanic | 0.15 | 0.21 | 0.128 |

White | 0.31 | 0.27 | 0.403 |

FRPL | 0.64 | 0.66 | 0.572 |

SPED | 0.06 | 0.05 | 0.875 |

LEP | 0.11 | 0.21 | 0.016 |

Prior achievement on state math test | 0.30 | 0.26 | 0.689 |

Prior achievement on state reading test | 0.28 | 0.30 | 0.782 |

p-value on joint test | 0.146 | ||

Teacher characteristics | |||

Prior teacher effects on state math scores | −0.01 | −0.01 | 0.828 |

Students | 67 | 531 |

*Note:* Means and *p*-values are calculated from regression framework that controls for randomization block. FRPL = free- or reduced-price lunch eligible; LEP = limited English proficiency status; SPED = special education status.

. | (1) . | (2) . | (3) . | (4) . | (5) . | (6) . | (7) . |
---|---|---|---|---|---|---|---|

Panel A: Unshrunken Estimates . | |||||||

State math test | 0.26 | NA | NA | 0.25 | 0.19 | 0.15 | 0.13 |

Behavior in Class | 0.48 | 0.38 | 0.38 | 0.41 | 0.31 | 0.29 | 0.16 |

Self-Efficacy in Math | 0.35 | 0.30 | 0.30 | 0.33 | 0.37 | 0.30 | 0.14 |

Happiness in Class | 0.48 | 0.45 | 0.44 | 0.43 | 0.43 | 0.34 | 0.27 |

Panel B: Shrunken Estimates | |||||||

State math test | 0.17 | NA | NA | 0.17 | 0.13 | 0.11 | 0.14 |

Behavior in Class | 0.33 | 0.22 | 0.22 | 0.26 | 0.19 | 0.21 | 0.10 |

Self-Efficacy in Math | 0.15 | 0.14 | 0.14 | 0.13 | 0.00 | 0.00 | 0.00 |

Happiness in Class | 0.34 | 0.31 | 0.31 | 0.34 | 0.34 | 0.30 | 0.33 |

Prior achievement | X | X | X | X | X | X | |

Prior survey response | X | X | |||||

Student characteristics | X | X | X | X | |||

Class characteristics | X | X | X | ||||

School characteristics | X | ||||||

School fixed effects | X | ||||||

Teachers | 51 | 51 | 51 | 51 | 51 | 51 | 51 |

Students | 548 | 548 | 548 | 548 | 548 | 548 | 548 |

. | (1) . | (2) . | (3) . | (4) . | (5) . | (6) . | (7) . |
---|---|---|---|---|---|---|---|

Panel A: Unshrunken Estimates . | |||||||

State math test | 0.26 | NA | NA | 0.25 | 0.19 | 0.15 | 0.13 |

Behavior in Class | 0.48 | 0.38 | 0.38 | 0.41 | 0.31 | 0.29 | 0.16 |

Self-Efficacy in Math | 0.35 | 0.30 | 0.30 | 0.33 | 0.37 | 0.30 | 0.14 |

Happiness in Class | 0.48 | 0.45 | 0.44 | 0.43 | 0.43 | 0.34 | 0.27 |

Panel B: Shrunken Estimates | |||||||

State math test | 0.17 | NA | NA | 0.17 | 0.13 | 0.11 | 0.14 |

Behavior in Class | 0.33 | 0.22 | 0.22 | 0.26 | 0.19 | 0.21 | 0.10 |

Self-Efficacy in Math | 0.15 | 0.14 | 0.14 | 0.13 | 0.00 | 0.00 | 0.00 |

Happiness in Class | 0.34 | 0.31 | 0.31 | 0.34 | 0.34 | 0.30 | 0.33 |

Prior achievement | X | X | X | X | X | X | |

Prior survey response | X | X | |||||

Student characteristics | X | X | X | X | |||

Class characteristics | X | X | X | ||||

School characteristics | X | ||||||

School fixed effects | X | ||||||

Teachers | 51 | 51 | 51 | 51 | 51 | 51 | 51 |

Students | 548 | 548 | 548 | 548 | 548 | 548 | 548 |

. | Behavior in Class . | Self-Efficacy in Math . | Happiness in Class . | |||
---|---|---|---|---|---|---|

. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. |

Panel A: Unshrunken Estimates . | ||||||

Teacher effects calculated from model 2 | 0.692^{***} | 0.059 | 0.452^{*} | 0.008 | 0.346^{*} | 0.000 |

(0.159) | (0.197) | (0.141) | ||||

Teacher effects calculated from model 3 | 0.702^{***} | 0.078 | 0.447^{*} | 0.013 | 0.347^{*} | 0.000 |

(0.165) | (0.213) | (0.141) | ||||

Panel B: Shrunken Estimates | ||||||

Teacher effects calculated from model 2 | 1.067^{***} | 0.805 | 0.604 | 0.279 | 0.423^{*} | 0.002 |

(0.269) | (0.360) | (0.175) | ||||

Teacher effects calculated from model 3 | 1.073^{***} | 0.799 | 0.563 | 0.241 | 0.421^{*} | 0.002 |

(0.286) | (0.367) | (0.175) | ||||

Teachers | 41 | 41 | 40 | |||

Students | 531 | 531 | 509 |

. | Behavior in Class . | Self-Efficacy in Math . | Happiness in Class . | |||
---|---|---|---|---|---|---|

. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. | Estimate/SE . | p-value on Difference from 1 SD
. |

Panel A: Unshrunken Estimates . | ||||||

Teacher effects calculated from model 2 | 0.692^{***} | 0.059 | 0.452^{*} | 0.008 | 0.346^{*} | 0.000 |

(0.159) | (0.197) | (0.141) | ||||

Teacher effects calculated from model 3 | 0.702^{***} | 0.078 | 0.447^{*} | 0.013 | 0.347^{*} | 0.000 |

(0.165) | (0.213) | (0.141) | ||||

Panel B: Shrunken Estimates | ||||||

Teacher effects calculated from model 2 | 1.067^{***} | 0.805 | 0.604 | 0.279 | 0.423^{*} | 0.002 |

(0.269) | (0.360) | (0.175) | ||||

Teacher effects calculated from model 3 | 1.073^{***} | 0.799 | 0.563 | 0.241 | 0.421^{*} | 0.002 |

(0.286) | (0.367) | (0.175) | ||||

Teachers | 41 | 41 | 40 | |||

Students | 531 | 531 | 509 |

*Notes:* Cells include estimates from separate regression models that control for students’ prior achievement in math and reading, student demographic characteristics, classroom characteristics from randomly assigned rosters, and fixed effects for randomization block. Robust standard errors clustered at the class level in parentheses. Model 2 calculates nonexperimental teacher effects controlling for a prior measure of students’ attitude or behavior. Model 3 controls for prior scores on both prior achievement and prior attitude or behavior. Both models include an indicator for whether or not students' lagged survey response was imputed.

^{*}*p* < 0.05, ^{***}*p* < 0.001.