8  Scale Coarseness

8.1 Introduction

Scale coarseness describes a situation where a variable that is naturally continuous (e.g., happiness) is binned into discrete values (e.g., happiness measured on a scale of 1-10). This situation is quite common in the social and psychological sciences where Likert items or dichotomous yes/no responses are aggregated to form a coarse total score for a naturally continuous construct. When coarseness is present, measurement error is introduced into the observed scores and those scores lose information.

8.2 Dichotomizing Continuous Random Variables

Unlike dichotomization, coarseness is an artifact that occurs due to the design of the study rather than during the analysis phase (Aguinis, Pierce, and Culpepper 2009). Particularly, dichotomization occurs after scores are obtained (e.g., splitting a group into high scorers and low scorers), whereas coarseness occurs as an artifact of the measurement procedure itself. This can be visualized by correlating coarse scores with their underlying continuous scores (see Figure 8.1). You will notice that the correlation between coarse and continuous scores is not perfect, indicating that the coarse scores do not perfectly capture the underlying continuous scores.

Figure 8.1: Scatterplot showing the correlation between coarse scores (on a 5-point scale) and the underlying continuous scores.

8.3 Correcting Correlations

8.3.1 Defining our Target Correlation

Our quantity of interest is the population correlation, \(\rho\), between continuous independent variable, \(X\), and continuous dependent variable, \(Y\). However, in a given study the measurement procedure may produce coarse scores for \(X\) and \(Y\). We will denote coarse scores with the subscript \(C\) We can model the relationship between the observed sample correlation on coarse scores and the true population correlation,

\[ r_{\widetilde{X}\widetilde{Y}} = \alpha\rho_{XY}+\varepsilon_r. \]

Where \(\alpha\) is our artifact attenuation factor and \(\varepsilon_r\) is our sampling error term.

8.3.2 Artifact Correction for Coarseness

Provided that the cuts are equally spaced, course attenuation of the correlation (MacCallum et al. 2002). Furthermore if we correlate a coarse score with another coarse score than we will observe even more attenuation (see Figure 8.2). There are two cases that we can run into: 1) the univariate case where only one variable is coarse and 2) the bivariate case where both variables are coarse.

Figure 8.2: First plot (left to right) shows both variables as continuous and normal. The second plot shows coarseness (5-point scale) only on X, leaving Y continuous (\(r_{\widetilde{X}Y}=.47\)). Last plot shows coarseness on both variables (\(r_{\widetilde{X}\widetilde{Y}}=.47\)).

To correct the correlation between coarse scores, we need to know the correlation between coarse scores and their underlying continuous scores (\(\rho_{X\widetilde{X}}\) or \(\rho_{Y\widetilde{Y}}\)). The calculation of the correlation will require us two important assumptions:

  1. The shape of the underlying distribution (i.e., normal or uniform).
  2. The intervals between scale-points are equal.

Based on these assumptions, Peters and Voorhis (1940) constructed a table of correlations between coarse and continuous scores that is also reported more recently by Aguinis, Pierce, and Culpepper (2009). Table 8.1 is adapted from Peters and Voorhis (1940) and displays the correlation values for uniform and normal distributions for a given number of scale points. For the normal distribution correction, its been shown that even in cases of extreme skew, these correction factors perform well (Wylie 1976).

Table 8.1: Correlations between continuous and coarse scores (\(\rho_{X\widetilde{X}}\)) from Peters and Voorhis (1940)
Scale Points Continuous-Coarse score Correlation (normal) Continuous-Coarse score Correlation (uniform)
2 .816 .866
3 .859 .943
4 .916 .968
5 .943 .980
6 .960 .986
7 .970 .990
8 .977 .992
9 .982 .994
10 .985 .995
11 .988 .996
12 .990 .997
13 .991 .997
14 .992 .997
15 .994 .998

The correlations between coarse and continuous scores from Table 8.1 (\(\rho_{X\widetilde{X}}\) and \(\rho_{Y\widetilde{Y}}\)) can be used as the \(\alpha\) to correct the correlation coefficient,

\[ r_{XY} = \frac{r_{\widetilde{X}\widetilde{Y}}}{\alpha} =\frac{r_{\widetilde{X}\widetilde{Y}}}{\rho_{X\widetilde{X}}\rho_{Y\widetilde{Y}}}. \]

Notice that provided that the assumptions are true, \(\alpha\) is known (the simulations used to compute \(\alpha\) can be arbitrarily precise with more iterations). Since \(\alpha\) is known we can easily,

\[ \widehat{\mathrm{var}}(r_{XY}) = \frac{\widehat{\mathrm{var}}(r_{\widetilde{X}\widetilde{Y}})}{\alpha^2} =\frac{\widehat{\mathrm{var}}(r_{\widetilde{X}\widetilde{Y}})}{\rho^2_{X\widetilde{X}}\rho^2_{Y\widetilde{Y}}} . \]

Correcting Correlations in R

Imagine that a researcher wants to relate depression and age. They collect a sample of 100 people from the general population and administer a very quick survey. Depression is measured based on a single item from the patient health questionnaire (PHQ, Kroenke, Spitzer, and Williams 2003) and age is measured in 5 age ranges:

Over the last 2 weeks, how often have you been bothered by feeling down, depressed, or hopeless?

What is your age?

  1. Not at all
  2. Several days
  3. More than half the days
  4. Nearly every day
  1. 1-20 years
  2. 21-40 years
  3. 41-60 years
  4. 61-80 years
  5. 81+ years

Let’s say we obtain a correlation of \(r_{\widetilde{X}\widetilde{Y}}=-.20\). Since the correlation is computed on coarse scores, it is likely attenuated relative to the correlation on each variables continuous underlying scores. Therefore we can use the correct_r_coarseness() function in the psychmeta package (Dahlke and Wiernik 2019) to correct the correlation.

library(psychmeta)

correct_r_coarseness(r = -.20, # observed correlation
                     kx = 5, # 5 age range bins
                     ky = 4,  # 4 PHQ item options
                     n = 100, # sample size
                     dist_x = "unif", # assumed age distribution
                     dist_y = "norm") # assumed depression distribution
  r_corrected var_e_corrected    n_adj
1  -0.2230339      0.01157681 78.99958

We see a slight increase in the magnitude of the correlation with the estimated correlation on continuous scores being \(r_{XY}=-.22\) \((\widehat{\mathrm{var}}(r_{XY})=.012)\).

8.4 Correcting Standardized Mean Differences (SMDs)

8.4.1 Defining our Target SMD

Our quantity of interest is the population SMD, \(\delta_{GY}\), between groups 0 and 1 on continuous variable, \(Y\). We can define the SMD on coarse scores (\(d_{G\widetilde{Y}}\)) as,

\[ d_{G\widetilde{Y}} = \alpha\delta_{GY}+\varepsilon_d. \]

Where \(\alpha\) is our coarseness biasing factor and \(\varepsilon_d\) is our sampling error term.

8.4.2 Artifact Correction for Coarseness

To correct a SMD for coarseness in dependent variable, we can use the correlation between coarse scores and continuous scores from Table 8.1,

\[ d_{GY} = \frac{d_{G\widetilde{Y}}}{\alpha} = \frac{d_{G\widetilde{Y}}}{\rho_{Y\widetilde{Y}}}. \]

We must also adjust sampling variance is,

\[ \widehat{\mathrm{var}}(d_{GY}) = \frac{\widehat{\mathrm{var}}(d_{G\widetilde{Y}}) }{\rho_{Y\widetilde{Y}}}. \]

Applied Example in R

Let’s say that a researcher wants to investigate gender differences in depressive symptoms. The researcher administers a survey to a sample of 50 men and 50 women from the general population. Depression is measured based on a single item from the patient health questionnaire (PHQ, Kroenke, Spitzer, and Williams 2003):

Over the last 2 weeks, how often have you been bothered by feeling down, depressed, or hopeless?

  1. Not at all
  2. Several days
  3. More than half the days
  4. Nearly every day

Let’s say we obtain a SMD of \(d_{G\widetilde{Y}}=.25\), slightly favoring women. Since there is currently no correct_d_coarseness() function in psychmeta, we can simply correct the correlation with base R:

library(psychmeta)

dGY <- .25 # SMD on coarse scores
var_d <- var_error_d(dGY,n1 = 50, n2 = 100) # SMD on coarse scores
rhoYY <- .916 # from table 10.1 (normal, 4-points)

dGY_corrected <- dGY / rhoYY # correct d
var_dGY_corrected <- var_d / rhoYY^2 # correct sampling variance

# print results
cbind(dGY_corrected, var_dGY_corrected)
     dGY_corrected var_dGY_corrected
[1,]     0.2729258        0.03649008

We see a slight increase in the magnitude of the correlation with the estimated correlation on continuous scores being \(d_{GY}=.27\). The adjusted sampling variance is \(\widehat{\mathrm{var}}(d_{GY})=.036\).