Many of the independent variables included in the analyses in this report are related, and to some extent the pattern of differences found in the descriptive analyses reflect this covariation. For example, when examining the proportion of faculty who taught classes for credit to undergraduates, it is impossible to know to what extent the observed variation is due to employment status differences and to what extent it is due to differences in other factors related to employment status, such as type of institution, academic rank held, and so on. However, if nested tables were used to isolate the influence of these other factors, cell sizes would become too small to identify the significant differences in patterns. When the sample size becomes too small to support controls for another level of variation, one must use other methods to take such variation into account. The method used in this report estimates adjusted means with regression models, an approach sometimes referred to as communality analysis.

Multiple linear regression was used to obtain means that were adjusted for covariation among a list of control variables.¹³ Each independent variable is divided into several discrete categories. To find an estimated mean value on the dependent variable for each category of an independent variable, while adjusting for its covariation with other independent variables in the equation, substitute the following in the equation: (1) a one in the category's term in the equation, (2) zeroes for the other categories of this variable, and (3) the mean proportions for all other independent variables. This procedure holds the impact of all remaining independent variables constant, and differences between adjusted means of categories of an independent variable represent hypothetical groups that are balanced or proportionately equal on all other characteristics included in the model as independent variables.

For example, consider a hypothetical case in which two variables, gender and employment status, are used to describe an outcome, Y (such as whether or not teaching classes for credit to undergraduates). The variables gender and employment status are recoded into dummy variables:

The following regression equation is then estimated from the correlation matrix output from the DAS:

To estimate the adjusted mean for any subgroup evaluated at the mean of all other variables, one substitutes the appropriate values for that subgroup's dummy variables (1 or 0) and the mean for the dummy variable(s) representing all other subgroups. For example, suppose we had a case where Y was being described by gender (G) and employment status (E), coded as shown above, and the means for G and E are as follows:

Suppose the regression equation results in:

To estimate the adjusted value for female faculty members, one substitutes the appropriate parameter values into equation 4.

This results in:

In this case, the adjusted proportion for female faculty is 0.719 and represents the expected outcome for the expected likelihood of teaching undergraduate classes for female faculty who look like average faculty with respect to the other variables in the model (in this example, employment status). In other words, the adjusted percentage of female faculty with tenure, controlling for employment status, is 71.9 percent (0.719 x 100 for conversion to a percentage). In addition to presenting the regression coefficients, their standard errors, and the unadjusted and adjusted percentages for each subgroup, the table of regression results also indicates the multiple R², the proportion of the variance in the outcome variable accounted for by all of the variables included in the multivariate model.

It is relatively straightforward to produce a multivariate model using the DAS, since one of the DAS output options is a correlation matrix, computed using pairwise missing values. In regression analysis, there are several common approaches to the problem of missing data. The two simplest are pairwise deletion of missing data and listwise deletion of missing data. In pairwise deletion, each correlation is calculated using all of the cases for the two relevant variables. For example, suppose you have a regression analysis that uses variables X1, X2, and X3. The regression is based on the correlation matrix between X1, X2, and X3. In pairwise deletion the correlation between X1 and X2 is based on the nonmissing cases for X1 and X2. Cases missing on either X1 or X2 would be excluded from the calculation of the correlation. In listwise deletion the correlation between X1 and X2 would be based on the nonmissing values for X1, X2, and X3. That is, all of the cases with missing data on any of the three variables would be excluded from the analysis.¹⁴

The correlation matrix can be used by most statistical software packages as the input data for least squares regression. That is the approach used for this report, with an additional adjustment to incorporate the complex sample design into the statistical significance tests of the parameter estimates (described below). For tabular presentation, parameter estimates and standard errors were multiplied by 100 to match the scale used for reporting unadjusted and adjusted percentages.

Most statistical software packages assume simple random sampling when computing standard errors of parameter estimates. Because of the complex sampling design used for the NSOPF survey, this assumption is incorrect. A better approximation of their standard errors is to multiply each standard error by the design effect associated with the dependent variable (DEFT),¹⁵ where the DEFT is the ratio of the true standard error to the standard error computed under the assumption of simple random sampling. It is calculated by the DAS and produced with the correlation matrix output.