- 1. The validity of inferences we draw from an experiment depend on the validity of the measures used.
- 2. Measurement is the link between a researcher’s substantive and/or theoretical argument and an (experimental) research design.
- 3. Measuring treatments includes the operationalization of treatment as well as compliance with treatment assignment.
- 4. Most outcomes of interest in social science are latent.
- 5. There are two types of measurement error that we should consider.
- 6. Measurement error reduces the power of your experiment.
- 7. Systematic measurement error biases estimates of causal effects of interests.
- 8. Leverage multiple indicators to assess the validity of a measure but be aware of the limitations of such tests.
- 9. The use of multiple indicators often improve the power of your experiment, but may introduce a bias-efficiency tradeoff.
- 10. While concepts may be global, many indicators are context-specific.
- Bibliography

We typically experiment in order to estimate the causal effect of a treatment, \(Z\), on an outcome, \(Y\). Yet, the reason that we care about estimating this causal effect is, in principle, to understand characteristics of the relationship between two theoretical, unobserved, concepts measured by observed variables \(Z\) and \(Y\).

Following Adcock and Collier (2001), consider the measurement process graphed in Figure 1 in three steps. First, researchers begin with systematized concept, a clearly-defined theoretical construct. From this concept, the researcher develops an indicator mapping the concept onto a scale or a set of categories. Finally, units or cases are scored on the indicator, yielding a measurement of treatment, \(Z\) and an outcome \(Y\). A measurement is valid if variation in the indicator closely approximates variation in the underlying concept of interest.

An experimental research design should allow a researcher to estimate the causal effect of \(Z\) on \(Y\) under standard assumptions. But if the ultimate goal is to make an inference about causal effect of the concept that \(Z\) measures on the concept that \(Y\) measures, the inferences that we can hope to make on the basis of our experimental evidence are valid if and only if both measures are valid.

When we consider the design of an experiment, we tend to focus on the process by which the randomly assigned treatment, \(Z\) is assigned and the joint distribution of \(Z\) and an outcome, \(Y\). In other words, we tend to divorce scores of \(Z\) and \(Y\) from broader concepts when considering the statistical properties of a research design. In this telling, two completely separate experiments with the same distribution of \(Z\) and \(Y\) could have identical properties.

For example, a clinical trial on the efficacy of aspirin on headaches and an experiment that provides information on an incumbent politician’s level of corruption and then asks the respondent if she will vote for the incumbent *could* have identical sized and distributed samples, assignments, estimands, and realizations of outcomes (data). Yet, this characterization of two completely distinct research projects that seek to make completely distinct inferences as “equivalent” may strike us as quite strange or even unsettling.

However, when we consider measurement as a fundamental component of research design, clearly these experiments are distinct. We observe measures of different concepts in the data for the two experiments. By considering the indicators and the broader concepts underlying the treatments and outcomes, we are forced to examine the researchers’ respective theories or arguments. In so doing, we can raise questions about the validity of the measures and the relationship between the validity of the measures and the validity of final, substantive, inferences.

In an experiment, treatments are typically designed, or at a minimum, described, by the researcher. Consumers of experimental research should be interested the characteristics of the treatment and how it manipulates a concept of interest. Most treatments in social science are compound, or include a bundle of attributes. We may be interested in the effect of providing voters with information on their elected officials’ performance. Yet, providing information also includes the mode of delivery and who was delivering the information. To understand the degree to which the treatment manipulates a concept, we must also understand what else the treatment could be manipulating.

However, despite all the effort operationalizing a treatment, in experimental research, the link from the operationalization to the treatment indicator is fundamentally is distinct from measurement of covariates or outcomes for two reasons. First, by assigning treatment, experimenters aim to control the values a given unit takes on. Second, for the treatment indicator, the score comes from assignment to treatment, which is a product of the randomization randomization. A subject may or may not have received the treatment, but her score on the treatment indicator is simply the treatment that she was assigned to, not the treatment she received.

When subjects receive different treatments than those to which they are assigned, we typically seek to measure compliance — whether the treatments were delivered and to what extent. To do so, we define what constitutes compliance with treatment assignment. In considering what constitutes compliance, researchers should consider the core aspect of how the treatment manipulates the concept of interest. At what point in the administration of the treatment does this manipulation occur? Once compliance is operationalized,we seek to code the compliance indicator in a manner faithful to this definition.

For example, consider a door-to-door canvassing campaign that distributes information about the performance of an incumbent politician. Households are assigned to receive a visit from canvasser that shares the information (treatment) or no visit (control). The treatment indicator is simply whether a household was assigned to the treatment or not. However, if residents of a household are not home when the canvasser visits, they do not receive the information. Our definition of compliance should determine what constitutes “treated” on our (endogenous) measure of whether a household received the treatment, here the information. Some common definitions of compliance may be (a) that someone from the household answered the door; or (b) that someone from the household listened to the full information script.

Many outcomes of interest – in experimental and non-experimental studies – are latent, or not directly observable. In social science, preferences, attitudes, and knowledge are common latent traits that we seek to explain. Yet, developing and validating measures for these phenomena presents difficulties.

We can formalize the measurement challenges quite simply. Suppose a treatment, \(Z_i\) is hypothesized to change preferences for democratic norms, \(\nu_i\). In principle, the quantity that we would like to estimate is \(E[\nu_i|Z_i = 1] - E[\nu_i|Z_i =0]\), the ATE of our treatment on preferences for democratic norms. However, \(\nu_i\) is a latent variable: we cannot measure it directly. Instead we ask about support for various behaviors thought to correspond to these norms. This indicator, \(Y_i\), can be decomposed into the latent variable, \(\nu_i\) and two forms of measurement error:

*Non-systematic measurement error*, \(\delta_i\): This error is independent of treatment assignment, \(\delta_i \perp Z_i\).*Systematic measurement error*, \(\kappa_i\): This error is not independent of treatment assigniment, \(\kappa_i \not\perp Z_i\).

\[Y_i = \underbrace{\nu_i}_{\text{Latent outcome}} + \underbrace{\delta_i}_{\substack{\text{Non-systematic} \\ \text{measurement error}}} + \underbrace{\kappa_i}_{\substack{\text{Systematic} \\ \text{measurement error}}}\]

Non-systematic measurement error, represented by \(\delta_i\) above, refers to the noise with which we are measuring the latent variable. In the absence of systematic measurement error, we measure:

\[Y_i = \underbrace{\nu_i}_{\text{Latent outcome}} + \underbrace{\delta_i}_{\substack{\text{Non-systematic} \\ \text{measurement error}}}\]

Now, consider the analytical power formula for a two-armed experiment. We can express \(\sigma\), or the standard deviation of the outcome as \(\sqrt{Var(Y_i)}\). Note that in the formula below, this term appears in the denominator of the first term. As \(\sqrt{Var(Y_i)}\) increases, statistical power decreases.

\[\beta = \Phi \left(\frac{|\mu_t− \mu_c| \sqrt{N}}{2 \color{red}{\sqrt{Var(Y_i)}}} − \Phi^{−1}\left(1 − \frac{\alpha}{2}\right)\right)\]

In what way does non-systematic measurement error \(\delta_i\) impact power? We can decompose \(\sqrt{Var(Y_i)}\) as follows:

\[\sqrt{Var(Y_i)} = \sqrt{Var(\nu_i) + Var(\delta_i) + 2 Cov(\nu_i, \delta_i)}\]

So long as \(Cov(\nu_i, \delta_i)\geq 0\) (we often assume \(Cov(\nu_i, \delta_i)= 0\)), it must be the case that the \(Var(Y_i)\) is increasing as measurement error, or \(Var(\delta_i)\) increases. This implies that power is decreasing as non-systematic measurement error increases. In other words, the noisier our measures of a latent variable, the lower our ability to detect effects of a treatment on a latent variable.

What about the case in which \(Cov(\nu_i, \delta_i) < 0\)? While this reduces \(Var(Y_i)\) (holding \(Var(\nu_i)\) and \(Var(\delta_i)\) constant), it also attenuates the variation that we measure in \(Y_i\). In principle, this should attenuate the numerator \(|\mu_t-\mu_c|\), which, if sufficient relative to the reduction in variance will also reduce power.

If we are estimating the Average Treatment Effect (ATE) of our treatment \(Z_i\), on preferences for democratic norms, \(\nu_i\), we are trying to recover the the ATE, or \(E[\nu_i|Z_i = 1] - E[\nu_i|Z_i =0]\). However, in the presence of **systematic** measurement error, where measurement error is related to the treatment assignment itself (say, the outcome is measured differently in the treatment group than in the control group) a difference-in-means estimator on the observed outcome, \(Y_i\), recovers a biased estimate of the ATE. The effect of the treatment now includes the measurement difference as well as the difference between treated and control groups:

\[E[Y_i|Z_i = 1]−E[Y_i|Z_i = 0] = E[\nu_i + \delta_i + \kappa_i |Z_i = 1] − E[\nu_i + \delta_i + \kappa_i|Z_i =0]\] Because non-systematic measurement error, \(\delta_i\) is independent of treatment assignment, \(E[\delta_i|Z_i = 1] = E[\delta_i |Z_i = 0]\). Simplifying and rearranging, we can write:

\[E[Y_i|Z_i = 1]−E[Y_i|Z_i = 0] = \underbrace{E[\nu_i|Z_i = 1] − E[\nu_i|Z_i =0]}_{ATE} + \underbrace{E[\kappa_i|Z_i = 1] - E[\kappa_i|Z_i =0]}_{\text{Bias}}\]

There are various sources of non-systematic measurement error in experiments. Demand effects and Hawthorne effects can be motivated as sources of systematic measurement error. Moreover, designs that measure outcomes asymmetrically in treatment and control groups may be prone to systematic measurement error. In all cases, there exists asymmetry across treatment conditions in: (a) the way that subjects respond to being observed; or (b) the way that we observe outcomes that is distinct from any effect of the treatment on the latent variable of interest. The biased estimate of the ATE becomes the net of any effects on the latent variables (the ATE) and the non-systematic measurement error.

Beyond consideration of the quality of the mapping between a concept and a measure, we can often assess the quality of the measure by comparing it to measures from alternate operationalizations of the same concept, closely related concepts, or distinct concepts. In *convergent tests* of the validity of a measure, we assess the correlation between alternate measures of a concept. If they are coded in the same direction, we expect the correlation to be positive and validity of both measures increases as the magnitude of the correlation increases. One limitation of convergent tests of validity is if two measures are weakly correlated, absent additional information, we do not know whether one measure is valid (and which) or whether both measures are invalid.

Gathering multiple indicators may also allow for researchers to assess the *predictive validity* of a measure. To what extent does a measure of a latent concept predict behavior believed to be shaped by the concept? For example, does political ideology (the latent variable) predict reported vote choice for left parties? This provides an additional means of validating a measure. Here, the higher the ability of an indicator to predict behavior (or other outcomes), the stronger the predictive validity of the indicator. Yet, we believe that most behaviors are a result of a complex array of causes. Determining whether a measure is a “good enough” predictor is a somewhat arbitrary determination.

Finally, we may want to determine whether we are measuring the concept of interest in isolation rather than a bundle of concepts. Tests of *discriminant validity* look at indicators of a concept and a related but distinct concept. In principle, we look for low correlations (correlations close to 0) between both indicators. One limitation of tests of *discriminant validity* is that we don’t know how underlying distinct concepts covary. It may be the case that we have valid indicators of both concepts, but they exhibit strong correlation (positive or negative) because units with high levels of \(A\) tend to have higher (resp. low) levels of \(B\).

In sum, the addition of more measures can help validate an indicator, but these validation tests are limited in what they tell us when they fail. To this extent, we should remain cognizant of the limitations in addition to the utility of collecting additional measures to simply to validate an indicator.

Gathering multiple indicators of a concept or outcome may also improve the power of your experiment. If multiple indicators measure the same concept but are measured with (non-systematic) error, we can improve the precision with which we measure the latent variable by leveraging multiple measures.

There are multiple ways to aggregate multiple outcomes into an index. “10 Things to Know about Multiple Comparisons” describes indices built from \(z\)-score and inverse covariance weighting of multiple outcomes. There are also many other structural models for estimating latent variables from multiple measures.

Below, we look at simple \(z\)-score index of two noisy measures of a latent variable. We assume that the latent variables and both indicators “Measure 1” and “Measure 2” are drawn from a multivariate normal distribution and are positively correlated with the latent variable and with each other. For the purposes of simulation, we assume that we know the latent variable, though in practice this is not possible. First, we can show that across many simulations of the data, the correlation between the \(z\)-score index of the two measures and the latent variable is, on average, higher than the correlation between either of the indicators and the latent variable. When graphing the correlation of the individual measures and the latent variable against (\(x\)-axes) the correlation of the index and the latent variable (\(y\)-axis), almost all points are above the 45-degree line. This shows that the index approximates the latent variable with greater precision.

```
library(mvtnorm)
library(randomizr)
library(dplyr)
library(estimatr)
make_Z_score <- function(data, outcome){
ctrl <- filter(data, Z == 0)
return(with(data, (data[,outcome] - mean(ctrl[,outcome]))/sd(ctrl[,outcome])))
}
pull_estimates <- function(model){
est <- unlist(model)$coefficients.Z
se <- unlist(model)$std.error.Z
return(c(est, se))
}
do_sim <- function(N, rhos, taus, var = c(1, 1, 1)){
measures <- rmvnorm(n = N,
sigma = matrix(c(var[1], rhos[1], rhos[2],
rhos[1], var[2], rhos[3],
rhos[2], rhos[3], var[3]), nrow = 3))
df <- data.frame(Z = complete_ra(N = N),
latent = measures[,1],
Y0_1 = measures[,2],
Y0_2 = measures[,3]) %>%
mutate(Yobs_1 = Y0_1 + Z * taus[1],
Yobs_2 = Y0_2 + Z * taus[2])
df$Ystd_1 = make_Z_score(data = df, outcome = "Yobs_1")
df$Ystd_2 = make_Z_score(data = df, outcome = "Yobs_2")
df$index = (df$Ystd_1 + df$Ystd_2)/2
cors <- c(cor(df$index, df$latent), cor(df$Ystd_1, df$latent), cor(df$Ystd_2, df$latent))
ests <- c(pull_estimates(lm_robust(Ystd_1 ~ Z, data = df)),
pull_estimates(lm_robust(Ystd_2 ~ Z, data = df)),
pull_estimates(lm_robust(index ~ Z, data = df)))
output <- c(cors, ests)
names(output) <- c("cor_index", "cor_Y1", "cor_Y2", "est_Y1", "se_Y1",
"est_Y2", "se_Y2", "est_index", "se_index")
return(output)
}
sims <- replicate(n = 500, expr = do_sim(N = 200,
rhos = c(.6, .6, .6),
taus = c(.4, .4),
var = c(1, 3, 3)))
data.frame(measures = c(sims["cor_Y1",], sims["cor_Y2",]),
index = rep(sims["cor_index",], 2),
variable = rep(c("Measure 1", "Measure 2"), each = 500)) %>%
ggplot(aes(x = measures, y = index)) + geom_point() +
facet_wrap(~variable) +
geom_abline(a = 0, b = 1, col = "red", lwd = 1.25) +
scale_x_continuous("Correlation between measure and latent variable", limits = c(0.1, .6)) +
scale_y_continuous("Correlation between index and latent variable", limits = c(0.1, .6)) +
theme_minimal()
```