The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Conducting post-hoc tests of compound coefficients using simple slopes for a categorical by continuous interaction Jane E. Miller, PhD The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

Overview Simple slopes defined Application of simple slopes to interactions Calculation of standard errors for simple slopes Charts to show conclusions of simple slope tests for interactions The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

Post-hoc probing Post-hoc (“after the fact”) probing is a way to conduct formal statistical tests of differences that were not specified for a priori inferential testing, e.g., – Comparisons other than against the reference category See podcast on testing statistical significance of differences across coefficients (  s) – Those involving more than one , e.g., interactions Can be conducted from output that is easily obtained from standard software packages Does not require respecifying the model

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Simple slopes calculation for compound coefficients The simple slopes technique is a way to calculate the point estimate of effect size for a combination of two coefficients from the same model The point estimate of effect size for each of those combinations is a compound coefficient – A linear combination of two  i s Models involving main effects and interaction terms require summing more than one  to obtain the overall effect of the variables involved in the interaction

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Review: Calculation of overall effect of an interaction The predicted value of the dependent variable from a model with a two-way interaction between predictors X 1 and X 2 can be written as a function of the estimated coefficients β i : Y = β 0 + β 1 X 1 + β 2 X 2 + β 3 X 1_ X 2 Rearranging terms to group those that involve the focal predictor (X 1 ) yields: (β 0 + β 2 X 2 ) + (β 1 +β 3 X 2 ) X 1, simple intercept ώ 0 simple slope ώ 1

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Review: Focal and modifier variables in an interaction In a statistical interaction between two independent variables X 1 and X 2 – X 1 is sometimes referred to as the focal predictor – X 2 is called the moderator or modifier variable because it alters the association between X 1 and Y Identifying which of the IVs in an interaction is the modifier depends on the research question – E.g., race/ethnicity is specified as the modifier because it is hypothesized to change the association between education and birth weight

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Standard errors for simple slopes Both the simple intercept and the simple slope are compound coefficients – Each is a linear combination of two estimated  i s Simple slope: ώ 0 = β 0 + β 2 X 2 Simple intercept : ώ 1 = (β 1 +β 3 X 2 ) X 1 The inferential statistical tests for a compound coefficient require calculating the standard error of the simple slope using the estimated variances and covariances for those  i s s.e. (ώ 1 | X 2 ) = √ [var(β 1 ) + (2 × X 2 × cov(β 1,β 3 )) + (X 2 ) 2 × var(β 3 )]

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Calculating the standard error for a compound coefficient s.e. (ώ 1 | X 2 ) is the standard error of the simple slope ώ 1 s.e. (ώ 1 | X 2 ) = √ [var(β 1 ) + (2 × X 2 × cov(β 1 β 3 )) + (X 2 ) 2 × var(β 3 )] Where  var(  j ) and var(  k ) are the variances of  j and  k, respectively  cov(  j,  k ) is the covariance between  j and  k The variance-covariance matrix can be requested as part of a regression command Note that s.e. (ώ 1 | X 2 ) depends on the value of the moderator variable in the interaction, X 2

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Formulas to calculate difference in birth weight by race for selected values of IPR Income-to-poverty ratio (IPR) Difference in birth weight compared to NHW with IPR = 0.0 Non-Hispanic whiteNon-Hispanic black 0 = (0 × β IPR ) = 0= β NHB + (0 × (β IPR + β NHB_IPR )) = β NHB 1 = (1× β IPR )= β NHB + (1 × (β IPR + β NHB_IPR )) … 4 = (4 × β IPR )= β NHB + (4 × (β IPR + β NHB_IPR )) More than one β is involved in calculations of the overall effect for any subgroup that is not in the reference category for either IV in the interaction – In this case when race is non-Hispanic black and IPR > 0

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Example: Calculation of the standard error of the simple slope for one race_IPR category Variance-covariance matrix for the estimated coefficients (βs) β NHB β IPR_NHB β NHB β IPR_NHB – In this example,  NHB is X 1, coded  1 = non-Hispanic black  0 = other racial/ethnic groups  IPR is X 2, a continuous variable with values ranging from 0 to 5  E.g., when IPR = 3, s.e. (ώ 1 | IPR) = √ [var(β NHB ) + (2 × IPR × cov(β NHB β IPR_NHB )) + (IPR) 2 × var(β IPR_NHB )] s.e. (ώ 1 | IPR) = √ [ (2 × 3× (– )) + (3) 2 × ] = 25.8

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Predicted birth weight with 95% CI, by race and income-to-poverty ratio (IPR) Confidence intervals can be evaluated re whether they overlap this value: predicted birth weight for non- Hispanic whites at IPR = 0

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Which contrasts are possible with non-Hispanic white, IPR = 0 as the reference category Predicted birth weight (grams), by race/ethnicity and income-to-poverty ratio (IPR) IPR Non-Hispanic white Non-Hispanic black Mexican American 0.03,1062,9293, ,1182,9383, ,1302,9473, ,1412,9563, ,1532,9653,110 ………… 4.03,2003,0023,113 The circled cell is the reference category (Non-Hispanic white, mother’s education > HS) Yellow-shaded cells can be compared to the reference category based on the standard errors of the associated main effects terms alone Green-shaded cells can be compared to the reference category using standard errors calculated using the simple slope for a compound coefficient

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Ballpark assessment of other contrasts If there is substantial overlap between confidence intervals for two values, you can usually safely conclude that they are not statistically significantly different from one another – It is very unlikely that taking the covariance between the two βs into account when computing the standard error of their simple slopes would alter that conclusion If the confidence intervals do not overlap or overlap only slightly, formally test the statistical significance of that difference by respecifying the model with a different reference category – See the podcast on that topic

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Online calculator for simple slopes Preacher has created an online calculator to compute simple intercepts, simple slopes, and their standard errors from regression output – Coefficients (βs) – Variances and covariances of the βs – Values of the moderator variable for which to calculate standard errors of the simple slope – Degrees of freedom for the model Can also graph the shape of the interaction with confidence intervals, given values of the focal predictor and moderator variables See

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Presenting results of post-hoc tests In methods section, mention use of simple slopes technique to calculate standard errors for compound coefficients. – Provide citation to that method Create a table to present the  i s, standard errors, and goodness-of-fit statistics from the multivariate model Conduct post-hoc tests for your substantive hypotheses behind the scenes – Calculate the simple slope – Calculate the standard error of the simple slope Create a table or chart to report predicted values and confidence intervals calculated from  i s and standard errors – Use symbols or prose to convey which contrasts other that against the reference category are statistically significantly different from one another

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Summary Overall pattern of an interaction involves calculation of compound coefficients Standard errors for compound coefficients can be calculated using the simple slope technique based on variance-covariance matrix from regression output Comparison of predicted values of the dependent variable are against the reference category, with all continuous independent variables set to 0 – To conduct contrasts for other values of IVs in the interaction Respecify the model with different reference categories Use the all-interaction-dummies approach Conduct calculations behind the scenes, present the conclusions of those tests in the text The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Separate podcasts

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Suggested resources Cohen, Jacob, Patricia Cohen, Stephen G. West, and Leona S. Aiken Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3rd Edition. Florence, KY: Routledge, chapters 7, 8, and 9. Figueiras, Adolfo, Jose Maria Domenech-Massons, and Carmen Cadarso Regression Models: Calculating the Confidence Interval of Effects in the Presence of Interactions. Statistics in Medicine 17: 2099–2105. Preacher, Kristopher J., Patrick J. Curran, and Daniel J. Bauer “Computational Tools for Probing Interaction Effects in Multiple Linear Regression, Multilevel Modeling, and Latent Curve Analysis.” Journal of Educational and Behavioral Statistics.31: 437–448. Miller, J. E The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. University of Chicago Press, chapter 16. The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

Suggested online resources Preacher, Kristopher J “Probing Interactions in Multiple Linear Regression, Latent Curve Analysis, and Hierarchical Linear Modeling: Interactive Calculation Tools for Establishing Simple Intercepts, Simple Slopes, and Regions of Significance.” Available online at

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Suggested online resources, cont. Podcasts on – Calculating the overall shape of an interaction from OLS coefficients – Approaches to testing statistical significance of interactions – Using alternative reference categories to test statistical significance of interactions

The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. Suggested practice exercises Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition. – Question #5 in the problem set for chapter 16 – Suggested course extensions for chapter 16 “Reviewing” exercise #2 “Applying statistics and writing” exercise #2 The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.

Contact information Jane E. Miller, PhD Online materials available at The Chicago Guide to Writing about Multivariate Analysis, 2nd Edition.