Overview of categorical by continuous interactions Part II

Overview of categorical by continuous interactions: Part II: Variables, specifications, and calculations Jane E. Miller, Ph. D The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Continued from Part I • Part I covers – Definitions and concepts for interactions – Possible shapes of patterns for interactions between one categorical and one continuous independent variable The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Creating variables and specifying models to test for interactions involving continuous independent variables The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Interaction between a continuous and a categorical independent variable (IV) • Example: Race and income-to-poverty ratio. – Race is a 2 -category IV classified • non-Hispanic black (NHB), • non-Hispanic white (NHW, ) – IPR is a continuous variable calculated as annual family income (in $) divided by the Federal Poverty Level for a family of that size and age composition. • IPR ranges from 0 to more than 10 in this sample. • Federal Poverty Level for a family of 2 adults and 2 children in 2010 was about $22, 000 The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Independent variables: continuous by continuous interaction • Mother’s age at time of child’s birth, years – One continuous variable for the main effect: age • Family income to poverty ratio, in multiples of the Federal Poverty Level – One continuous variable for the main effect: IPR • Interaction: Mother’s age and IPR – Age_IPR = age × IPR – Resulting interaction term variable will also be continuous The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Model specification to test an interaction between one continuous and one categorical independent variable • For a model with an interaction between two independent variables, need all of the ALL of the main effects and interaction term variables related to those two independent variables. • E. g. , for a model of birth weight by race and IPR, include the main effect and interaction terms related to race and family IPR-to-poverty ratio: – BW = f (NHB, IPR, NHB_IPR) The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Coding of variables • The NHB main effect variable is defined as in the previous example (of categorical by categorical interaction). • 1 = non-Hispanic black. • 0 = all others, the reference category, in this example, non. Hispanic white. • However, for a continuous variable like income that takes on many possible numeric values, it doesn’t make sense to create a lot of dummy variables. • Instead, use income-poverty ratio in its continuous form. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculating an interaction term from a dummy and a continuous main effects term • The value of the interaction term variable is defined as the product of the two component main effects variables: X 1 _ X 2 = X 1 × X 2 – Result will be one continuous interaction term variable. • Thus NHB_IPR is the product of NHB and IPR. – If NHB = 1 and IPR = 2. 3 then the interaction term NHB_IPR = 2. 3 – If NHB = 0 and IPR = 2. 3, then NHB_IPR = 0 The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Coding of main effects and interaction term variables: race and IPR Case characteristics – SELECTED VALUES Non-H white & IPR = 0. 5 Non-H white & IPR = 1. 0 Non-H white & IPR = 2. 0 Non-H white & IPR = 5. 0 Non-H black & IPR = 0. 5 Non-H black & IPR = 1. 0 Non-H black & IPR = 2. 0 Non-H black & IPR = 5. 0 Variables Main effects terms Interaction term NHB IPR NHB_IPR 0 0 1 1 0. 5 1. 0 2. 0 5. 0 0 0 0. 5 1. 0 2. 0 5. 0 E. g. , IPR = 0. 5 means income is half the Federal Poverty Level (FPL); IPR = 2. 0 means income is twice the FPL. For a two-category race variable (non-Hispanic white = reference category). The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Coding of race and IPR variables: Non-Hispanic white infants Case characteristics Non-H white & IPR = 0. 5 Non-H white & IPR = 1. 0 Non-H white & IPR = 2. 0 Non-H white & IPR = 5. 0 Variables Main effects terms Interaction term NHB IPR NHB_IPR 0 0 0. 5 1. 0 2. 0 5. 0 0 0 E. g. , IPR = 0. 5 means income is half the Federal Poverty Level (FPL); IPR = 2. 0 means income is twice the FPL. For a two-category race variable (non-Hispanic white = reference category). The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Coding of race and IPR variables: Non-Hispanic black infants Case characteristics Non-H black & IPR = 0. 5 Non-H black & IPR = 1. 0 Non-H black & IPR = 2. 0 Non-H black & IPR = 5. 0 Variables Main effects terms Interaction term NHB IPR NHB_IPR 1 1 0. 5 1. 0 2. 0 5. 0 E. g. , IPR = 0. 5 means income is half the Federal Poverty Level (FPL); IPR = 2. 0 means income is twice the FPL. For a two-category race variable (non-Hispanic white = reference category). The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

General equation for predicted value of DV based on an interaction model • The general equation to calculate the predicted value of the dependent variable includes – main effects coefficients – interaction term coefficients – values of the independent variables = β 0 + (βNHB × NHB) + (βIPR × IPR) + (βNHB_IPR × NHB_IPR) The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculating overall effect of interaction for specific case characteristics = β 0 + (βNHB × NHB) + (βIPR × IPR) + (βNHB_IPR × NHB_IPR) • Each coefficient is multiplied by the value of the associated variable for cases with the characteristics of interest. • To see which coefficients pertain to which cases, fill in values of variables for different combinations of race and the income-to-poverty ratio (IPR). The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Example: Estimated coefficients β Intercept Main effect terms Non-Hispanic black (NHB) Income-to-poverty ratio (IPR) Interaction term NHB_IPR 3, 106 – 177 23 – 5 IPR = family income ($) / Federal Poverty Level for a family of that size and age composition. Reference category: Non-Hispanic whites. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Interpreting the intercept • The intercept β 0 from an OLS model is an estimate of the level of the dependent variable when continuous variables take the value 0, for infants in the reference category for all categorical variables. • In a model where – The dependent variable is birth weight in grams. – The reference category is specified to be non-Hispanic white infants. • β 0 is an estimate of birth weight when IPR = 0, for non. Hispanic white infants. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Review: Coding of main effect and interaction term variables: race and income Reference category Variables Case characteristics – Main effects terms Interaction term NHB IPR NHB_IPR SELECTED VALUES Non-H white & IPR = 0. 0 Non-H white & IPR = 0. 5 Non-H white & IPR = 1. 0 0 0. 5 1. 0 0 E. g. , IPR = 0. 5 means family income is half the Federal Poverty Level (FPL); IPR = 2. 0 means family income is twice the FPL. For a two-category race variable (non-Hispanic white = reference category). The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculating the value of the intercept for one group = β 0 + (βNHB × NHB) + (βIPR × IPR) + (βNHB_IPR × NHB_IPR) Non-H white & IPR = 0. 0 NHB 0 IPR 0. 0 NHB_IPR 0. 0 The intercept for non-Hispanic whites is calculated: = β 0 + (βNHB × 0) + (βIPR × 0. 0) + (βNHB_IPR × 0. 0) = β 0 Thus, the intercept for non-Hispanic white infants (when IPR = 0) collapses to include only β 0 because all of the other coefficients in the formula are multiplied by a value of 0. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Interpreting the IPR/birth weight pattern • IPR is a continuous variable – The coefficient is an estimate of the effect on the dependent for a 1 -unit increase in the continuous IV, with categorical variables set to their reference category values. • So βIPR estimates the increment in birth weight for every one-unit increase in IPR (e. g. , from family income at the poverty line to twice the poverty line) – It is the slope of the IPR/birth weight curve for infants in the reference category, in this case, non-Hispanic white infants. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculating values for the IPR/birth weight curve for white infants = β 0 + (βNHB × NHB) + (βIPR × IPR) + (βNHB_IPR × NHB_IPR) Non-H white & IPR = 1. 5 NHB 0 IPR 1. 5 NHB_IPR 0. 0 = β 0 + (βNHB × 0) + (βIPR × 1. 5) + (βNHB_IPR × 0) = β 0 + (βIPR × 1. 5) Because non-Hispanic whites are the reference category for race, the equation collapses to include only the IPR main effect (βIPR) because the other coefficients are multiplied by 0. = β 0 + (βIPR × IPR) The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Interpreting the race main effect • The main effect βNHB estimates the difference in birth weight between non-Hispanic black infants and those in the reference category (non-Hispanic whites), when continuous variables are set at the value 0. • It is an estimate of the difference in intercept between black and white infants when IPR is 0. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculating the intercept for different values of the categorical variable NHB IPR NHB_IPR Non-H white & IPR = 0. 0 As we saw a moment ago, for the intercept for non-Hispanic whites is calculated: = β 0 + (βNHB × 0) + (βIPR × 0. 0) + (βNHB_IPR × 0. 0) = β 0 Non-H black & IPR = 0. 0 NHB 1 IPR 0. 0 NHB_IPR 0. 0 For non-Hispanic blacks, the intercept is calculated: = β 0 + (βNHB × 1) + (βIPR × 0. 0) + (βNHB_IPR × 0. 0) = β 0 + βNHB The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

More on the race main effect • It is an estimate of the difference in intercept between black and white infants when IPR is 0. = β 0 + βNHB = 3, 106 + (– 177) = 2, 929 • In other words, black infants born to families with an IPR of zero have a predicted birth weight of 2, 929 grams. – or 177 grams LOWER than that of their white counterparts. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculating values for the IPR/birth weight curve for white infants = β 0 + (βNHB × NHB) + (βIPR × IPR) + (βNHB_IPR × NHB_IPR) = β 0 + (βNHB × 0) + (βIPR × IPR) + (βNHB_IPR × 0) = β 0 + (βIPR × IPR) Because non-Hispanic whites are the reference category for race, the equation collapses to include only the IPR main effect (βIPR) because the other coefficients are multiplied by 0. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculating values for the IPR birth weight curve for black infants Non-H black & IPR = 1. 5 NHB 1 IPR 1. 5 NHB_IPR 1. 5 = β 0 + (βNHB × 1) + (βIPR × 1. 5) + (βNHB_IPR × 1. 5) For Non-Hispanic blacks, the equation includes all three terms (βNHB, βIPR, and βNHB_IPR) because each of those coefficients is multiplied by a non-zero value. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Interpreting the coefficient on the interaction between race and IPR • The slope – for blacks = βIPR + βNHB_IPR = 23 + (– 5) = 18 – for whites = βIPR = 23 • The race_IPR coefficient tests whether the slope of the IPR/birth weight pattern is different for non-Hispanic black infants than for their non. Hispanic white counterparts. – βNHB_IPR is thus the estimated difference in slope for blacks compared to whites. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

More on the race/IPR interaction • The estimated coefficients mean that each 1 unit increase in IPR is associated with § 23 grams more birth weight among non-Hispanic white infants. § 18 grams more birth weight among non-Hispanic black infants. § Thos values are the slopes of the respective IPR/BW curves for the two racial/ethnic groups. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Preparing to graph the slope of IPR/birth weight by race • For infants in the reference category (non. Hispanic white), – Multiply selected values of IPR by βIPR and add to β 0 to obtain predicted birth weight at interesting values of IPR. • For non-Hispanic black infants, – Multiply selected values of IPR by βIPR + βNHB_IPR then add to β 0 + βNHB. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Calculated birth weight by race for selected values of IPR (family Non-Hispanic white income in multiples of the Formula Result FPL) 0 1 … 6 Non-Hispanic black = β 0 + 0 × βIPR = 3, 106 + 0× 23 = β 0 + 1× βIPR = 3, 106 + 1× 23 = 3, 106 + 23 Formula Result = β 0 + βNHB + 0 × (βIPR + βNHB_IPR) 3, 106 = 3, 106 – 177 + 0 × (23 – 5) 2, 929 = β 0 + βNHB + 1 × (βIPR + βNHB_IPR) = 3, 106 – 177 + 1 × (23 – 5) 3, 129 = 2, 929 + 1 × (18) = 2, 929 + 18 2, 947 = β 0 + 6 × βIPR = 3, 106 + 6× 23 = 3, 106 + 390 = β 0 + βNHB + 6 × (βIPR + βNHB_IPR) = 3, 106 – 177 + 6 × (23 – 5) 3, 244 = 2, 929 + 6 × (18) = 2, 929 + 108 3, 037 β 0 = 3, 106; βIPR = 23; βNHB = – 177; βNHB_IPR = – 5 The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Use a spreadsheet to calculate and graph the interaction • Spreadsheets can – Store • The estimated coefficients • The input values of the independent variables • The correct generalized formula to calculate the predicted values for many combinations of the IVs involved in the interaction – Graph the overall pattern • See spreadsheet template and voice-over explanation The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Predicted birth weight by race/ethnicity and IPR = βIPR = 23 = slope of IPR/ BW curve for ref cat * Birth weight (grams) 3, 300 3, 200 = β 0 = intercept = 3, 106 = predicted BW for ref cat * = βIPR + βNHB_IPR = 23 – 5 = 18 = slope of IPR/ BW curve for non-Hispanic black infants 3, 100 3, 000 2, 900 = β 0 + βNHB = 3, 106 + (– 177) = 2, 929 = intercept for black infants 2, 800 0 1 2 4 6 IPR * Ref cat = Reference category = non-Hispanic white infants. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Overall shape of the race/IPR/ birth weight pattern • Based on this set of βs, black infants have – a lower birth weight than whites at all IPR levels. • Negative coefficient on the NHB main effect yields a lower intercept for blacks than for whites. – a slower rate of birth weight increase as IPR rises. • Negative coefficient on NHB_IPR, which yields a shallower slope of the IPR/birth weight curve for blacks than for whites. • Thus the deficit in birth weight for blacks widens with increasing IPR. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Summary • An interaction between a continuous and a categorical independent variable will yield differences in the intercept and/or slope of the association between the continuous IV and the DV. • Calculating the overall shape of an interaction requires adding together the pertinent main effects and interaction term βs for combinations of the categorical IV and selected values of the continuous IV in the interaction. – A spreadsheet can be helpful for storing and organizing the βs, input values, and formulas. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Be parsimonious in deciding which interactions to test • The number of variables in the regression model proliferates rapidly with each additional interaction. • Specify interactions only between key independent variables. • Communicating results becomes unwieldy: – Considerable behind-the-scenes calculations. – Extra tables or charts to convey the shape of the interaction. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Suggested resources • Chapter 16, Miller, J. E. 2013. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition. • Chapters 8 and 9 of Cohen et al. 2003. Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences, 3 rd Edition. Florence, KY: Routledge. The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Suggested online resources • Podcasts on – Creating charts to present interactions – Writing prose to present results of interactions – Introduction to testing statistical significance of interactions – Approaches to testing statistical significance of interactions – Using simple slopes for compound coefficients – Using alternative reference categories to test contrasts within interactions The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Suggested exercises • Study guide to The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition. – Problem set for chapter 16 – Suggested course extensions for chapter 16 The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.

Contact information Jane E. Miller, Ph. D jmiller@ifh. rutgers. edu Online materials available at http: //press. uchicago. edu/books/miller/multivariate/index. html The Chicago Guide to Writing about Multivariate Analysis, 2 nd Edition.