Lesson 12: Interactions, Part 2

Nicky Wakim

2026-02-25

Learning Objectives

Last time:

  1. Define confounders and effect modifiers, and how they interact with the main relationship we model.
  2. Interpret the interaction component of a model with a binary categorical covariate and continuous covariate, and how the main variable’s effect changes.
  3. Interpret the interaction component of a model with a multi-level categorical covariate and continuous covariate, and how the main variable’s effect changes.

This time:

  1. Interpret the interaction component of a model with two categorical covariates, and how the main variable’s effect changes.
  2. Interpret the interaction component of a model with two continuous covariates, and how the main variable’s effect changes.
  3. Report results for a best-fit line (with confidence intervals) at different levels of an effect measure modifier

Learning Objectives

Last time:

  1. Define confounders and effect modifiers, and how they interact with the main relationship we model.
  2. Interpret the interaction component of a model with a binary categorical covariate and continuous covariate, and how the main variable’s effect changes.
  3. Interpret the interaction component of a model with a multi-level categorical covariate and continuous covariate, and how the main variable’s effect changes.

This time:

  1. Interpret the interaction component of a model with two categorical covariates, and how the main variable’s effect changes.
  1. Interpret the interaction component of a model with two continuous covariates, and how the main variable’s effect changes.
  2. Report results for a best-fit line (with confidence intervals) at different levels of an effect measure modifier

Do we think income level can be an effect modifier for freedom status?

  • Taking a break from cell phones to demonstrate interactions for two categorical variables

  • We can start by visualizing the relationship between life expectancy and freedom status by income level

  • Questions of interest: Does the effect of freedom status on life expectancy differ depending on income level?

    • This is the same as: Is income level an effect modifier for freedom status?

  • Let’s run an interaction model to see!

Model with interaction between a multi-level categorical and binary variables

Model we are fitting:

\[\begin{aligned}LE = &\beta_0 + \beta_1 I(\text{high income}) + \beta_2 I(\text{FS} = \text{PF}) + \beta_3 I(\text{FS} = \text{F}) + \\ & \beta_4 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + \beta_5 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{F}) + \epsilon \end{aligned}\]

  • \(LE\) as life expectancy
  • \(I(\text{high income})\) as indicator of high income
  • \(I(\text{FS} = \text{PF})\) and \(I(\text{FS} = \text{F})\) as the indicator for each freedom status

In R:

m_int_wr_inc = gapm %>% 
  lm(formula = life_exp ~ income_level_2 + freedom_status + income_level_2*freedom_status)
m_int_wr_inc = gapm %>% 
  lm(formula = life_exp ~ income_level_2*freedom_status)

Displaying the regression table and writing fitted regression equation

tidy_m_int_wr_inc = tidy(m_int_wr_inc, conf.int=T)
tidy_m_int_wr_inc %>% gt() %>% tab_options(table.font.size = 35) %>% fmt_number(decimals = 3)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 67.355 1.179 57.126 0.000 65.015 69.695
income_level_2Higher income 5.645 2.188 2.580 0.011 1.303 9.987
freedom_statusPF 0.029 1.588 0.018 0.985 −3.123 3.181
freedom_statusF −5.665 3.404 −1.664 0.099 −12.419 1.089
income_level_2Higher income:freedom_statusPF 1.633 2.744 0.595 0.553 −3.813 7.078
income_level_2Higher income:freedom_statusF 8.287 4.026 2.058 0.042 0.299 16.275

\[\begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 I(\text{high income}) + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) + \\ & \widehat\beta_4 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + \widehat\beta_5\cdot I(\text{high income}) \cdot I(\text{FS} = \text{F})\\ \widehat{LE} = & 67.35 + 5.64 \cdot I(\text{high income}) + 0.03 \cdot I(\text{FS} = \text{PF}) -5.67 \cdot I(\text{FS} = \text{F}) + \\ & 1.63 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + 8.29\cdot I(\text{high income}) \cdot I(\text{FS} = \text{F})\\ \end{aligned}\]

Poll Everywhere Question 4

Comparing fitted regression means for each freedom status

\[\begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 I(\text{high income}) + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) + \\ & \widehat\beta_4 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + \widehat\beta_5\cdot I(\text{high income}) \cdot I(\text{FS} = \text{F})\\ \widehat{LE} = & 67.35 + 5.64 \cdot I(\text{high income}) + 0.03 \cdot I(\text{FS} = \text{PF}) -5.67 \cdot I(\text{FS} = \text{F}) + \\ & 1.63 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + 8.29\cdot I(\text{high income}) \cdot I(\text{FS} = \text{F}) \end{aligned}\]

Not free

\[\begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 I(\text{high income}) + \\ & \widehat\beta_2 \cdot 0 + \widehat\beta_3 \cdot 0 + \\ & \widehat\beta_4 \cdot I(\text{high income}) \cdot 0 + \\ & \widehat\beta_5\cdot I(\text{high income}) \cdot 0 \\ \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 I(\text{high income}) \end{aligned}\]

Partly free

\[\begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 I(\text{high income}) + \\ & \widehat\beta_2 \cdot 1 + \widehat\beta_3 \cdot 0 + \\ & \widehat\beta_4 \cdot I(\text{high income}) \cdot 1 + \\ & \widehat\beta_5\cdot I(\text{high income}) \cdot 0 \\ \widehat{LE} = & \big(\widehat\beta_0 + \widehat\beta_2 \big) + \\ & \big(\widehat\beta_1 + \widehat\beta_4 \big) I(\text{high income}) \\ \end{aligned}\]

Free

\[\begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 I(\text{high income}) + \\ & \widehat\beta_2 \cdot 1 + \widehat\beta_3 \cdot 0 + \\ & \widehat\beta_4 \cdot I(\text{high income}) \cdot 1 + \\ & \widehat\beta_5\cdot I(\text{high income}) \cdot 0 \\ \widehat{LE} = & \big(\widehat\beta_0 + \widehat\beta_3 \big) + \\ & \big(\widehat\beta_1 + \widehat\beta_5 \big) I(\text{high income}) \\ \end{aligned}\]

Comparing fitted regression means for each income level

\[\begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 I(\text{high income}) + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) + \\ & \widehat\beta_4 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + \widehat\beta_5\cdot I(\text{high income}) \cdot I(\text{FS} = \text{F})\\ \widehat{LE} = & 67.35 + 5.64 \cdot I(\text{high income}) + 0.03 \cdot I(\text{FS} = \text{PF}) -5.67 \cdot I(\text{FS} = \text{F}) + \\ & 1.63 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + 8.29\cdot I(\text{high income}) \cdot I(\text{FS} = \text{F}) \end{aligned}\]

For lower income countries: \(I(\text{high income})=0\)

\[ \begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 \cdot 0 + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) + \\ & \widehat\beta_4 \cdot 0 \cdot I(\text{FS} = \text{PF}) + \widehat\beta_5 \cdot 0 \cdot I(\text{FS} = \text{F})\\ \widehat{LE} = & \widehat\beta_0 + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) \\ \end{aligned}\]

For higher income countries: \(I(\text{high income}) =1\)

\[ \begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 \cdot 1 + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) + \\ & \widehat\beta_4 \cdot 1 \cdot I(\text{FS} = \text{PF}) + \widehat\beta_5 \cdot 1 \cdot I(\text{FS} = \text{F})\\ \widehat{LE} = & \big(\widehat\beta_0 + \widehat\beta_1 \big) + \big(\widehat\beta_2 + \widehat\beta_4 \big) I(\text{FS} = \text{PF}) + \\ & \big(\widehat\beta_3 + \widehat\beta_5 \big) I(\text{FS} = \text{F}) \\ \end{aligned}\]

  • Example interpretation: The America’s effect on mean life expectancy increases \(\widehat{\beta}_5\) comparing high income to low income countries.

Let’s take a look back at the plot

For lower income countries: \(I(\text{high income}) =0\)

\[ \begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) \\ \end{aligned}\]

For higher income countries: \(I(\text{high income}) =1\)

\[ \begin{aligned} \widehat{LE} = & \big(\widehat\beta_0 + \widehat\beta_1 \big) + \big(\widehat\beta_2 + \widehat\beta_4 \big) I(\text{FS} = \text{PF}) + \\ & \big(\widehat\beta_3 + \widehat\beta_5 \big) I(\text{FS} = \text{F}) \\ \end{aligned}\]

Interpretation for interaction between two categorical variables

\[ \begin{aligned} \widehat{LE} = &\widehat\beta_0 + \widehat\beta_1 \cdot 1 + \widehat\beta_2 I(\text{FS} = \text{PF}) + \widehat\beta_3 I(\text{FS} = \text{F}) + \\ & \widehat\beta_4 \cdot 1 \cdot I(\text{FS} = \text{PF}) + \widehat\beta_5 \cdot 1 \cdot I(\text{FS} = \text{F})\\ \widehat{LE} = & \bigg[\widehat\beta_0 + \widehat\beta_1 \cdot I(\text{high income})\bigg] + \bigg[\widehat\beta_2 + \widehat\beta_4 \cdot I(\text{high income})\bigg] I(\text{FS} = \text{PF}) + \\ & \bigg[\widehat\beta_3 + \widehat\beta_5 \cdot I(\text{high income})\bigg] I(\text{FS} = \text{F}) \\ \end{aligned}\]

  • Interpretation:
    • \(\widehat\beta_1\) = mean change in life expectancy comparing high income to low income countries, for countries/territories that are not free
    • \(\widehat\beta_4\) = mean change in life expectancy comparing high income to low income countries, for countries/territories that are partly free
    • \(\widehat\beta_5\) = mean change in life expectancy comparing high income to low income countries, for countries/territories that are free

Test interaction between two categorical variables (1/2)

  • We run an F-test for a group of coefficients (\(\beta_4\), \(\beta_5\)) in the below model (see lesson 9)

\[\begin{aligned}LE = &\beta_0 + \beta_1 I(\text{high income}) + \beta_2 I(\text{FS} = \text{PF}) + \beta_3 I(\text{FS} = \text{F}) + \\ & \beta_4 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + \beta_5 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{F}) + \epsilon \end{aligned}\]

Null \(H_0\)

\(\beta_4= \beta_5 =0\)

Alternative \(H_1\)

\(\beta_4\neq0\) and/or \(\beta_5\neq0\)

Null / Smaller / Reduced model

\[\begin{aligned}LE = &\beta_0 + \beta_1 I(\text{high income}) + \\ &\beta_2 I(\text{FS} = \text{PF}) + \beta_3 I(\text{FS} = \text{F}) + \\ & \epsilon \end{aligned}\]

Alternative / Larger / Full model

\[\begin{aligned}LE = &\beta_0 + \beta_1 I(\text{high income}) + \beta_2 I(\text{FS} = \text{PF}) + \\ & \beta_3 I(\text{FS} = \text{F}) + \beta_4 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{PF}) + \\ & \beta_5 \cdot I(\text{high income}) \cdot I(\text{FS} = \text{F}) + \epsilon \end{aligned}\]

Test interaction between two categorical variables (2/2)

  • Fit the reduced and full model
m_int_wr_inc_red = gapm %>% 
  lm(formula = life_exp ~ income_level_2 + freedom_status)
m_int_wr_inc_full = gapm %>% 
  lm(formula = life_exp ~ income_level_2*freedom_status)
Display the ANOVA table with F-statistic and p-value 
anova_table = anova(m_int_wr_inc_red, m_int_wr_inc_full)
anova_table %>% tidy() %>% gt() %>% tab_options(table.font.size = 35) %>% 
  fmt_number(decimals = 3)
term df.residual rss df sumsq statistic p.value
life_exp ~ income_level_2 + freedom_status 101.000 3,160.771 NA NA NA NA
life_exp ~ income_level_2 * freedom_status 99.000 3,027.855 2.000 132.915 2.173 0.119
  • Conclusion: There is not a significant interaction between freedom status and income level (p = 0.119).

Learning Objectives

Last time:

  1. Define confounders and effect modifiers, and how they interact with the main relationship we model.
  2. Interpret the interaction component of a model with a binary categorical covariate and continuous covariate, and how the main variable’s effect changes.
  3. Interpret the interaction component of a model with a multi-level categorical covariate and continuous covariate, and how the main variable’s effect changes.

This time:

  1. Interpret the interaction component of a model with two categorical covariates, and how the main variable’s effect changes.
  1. Interpret the interaction component of a model with two continuous covariates, and how the main variable’s effect changes.
  1. Report results for a best-fit line (with confidence intervals) at different levels of an effect measure modifier

Do we think vaccination rate is an effect modifier for cell phones?

  • We can start by visualizing the relationship between life expectancy and cell phones by vaccination rate

  • Questions of interest: Does the effect of cell phones on life expectancy differ depending on vaccination rate?

    • This is the same as: Is vaccination rate is an effect modifier for cell phones? Is vaccination rate an effect modifier of the association between life expectancy and cell phones?

  • Let’s run an interaction model to see!

Model with interaction between two continuous variables

Model we are fitting:

\[ LE = \beta_0 + \beta_1 CP^c + \beta_2 VR^c + \beta_3 CP^c \cdot VR^c + \epsilon\]

  • \(LE\) as life expectancy
  • \(CP^c\) as the centered around the mean number of cell phones per 100 people (continuous variable)
  • \(VR^c\) as the centered around the mean vaccination rate (continuous variable)
Code to center CP and VR 
gapm = gapm %>% 
  mutate(CP_c = cell_phones_100 - mean(cell_phones_100), 
         VR_c = vax_rate - mean(vax_rate))
mean_VR = mean(gapm$vax_rate) %>% round(digits = 2)
mean_CP = mean(gapm$cell_phones_100) %>% round(digits = 2)

In R:

m_int_vr = gapm %>% lm(formula = life_exp ~ CP_c + VR_c + CP_c*VR_c)

OR

m_int_vr = gapm %>% lm(formula = life_exp ~ CP_c*VR_c)

Displaying the regression table and writing fitted regression equation

tidy_m_vr = tidy(m_int_vr, conf.int=T) 
tidy_m_vr %>% gt() %>% tab_options(table.font.size = 35) %>% fmt_number(decimals = 5)
term estimate std.error statistic p.value conf.low conf.high
(Intercept) 70.56972 0.61989 113.84216 0.00000 69.34002 71.79942
CP_c 0.07666 0.01832 4.18404 0.00006 0.04032 0.11301
VR_c 0.23765 0.08409 2.82598 0.00568 0.07083 0.40447
CP_c:VR_c 0.00308 0.00192 1.59912 0.11292 −0.00074 0.00689

\[ \begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP^c + \widehat\beta_2 VR^c + \widehat\beta_3 CP^c \cdot VR^c \\ \widehat{LE} = & 70.57 + 0.08 \cdot CP^c + 0.24 \cdot VR^c + 0.003 \cdot CP^c \cdot VR^c \end{aligned}\]

Comparing fitted regression lines for various vaccination rates

\[ \begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP^c + \widehat\beta_2 VR^c + \widehat\beta_3 CP^c \cdot VR^c \\ \widehat{LE} = & 70.57 + 0.08 \cdot CP^c + 0.24 \cdot VR^c + 0.003 \cdot CP^c \cdot VR^c \end{aligned}\]

To identify different lines, we need to pick example vaccination rates:

Vaccination rate of 86.45 %

\[\begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP^c + \\ & \widehat\beta_2 \cdot (-5) + \\ & \widehat\beta_3 CP^c \cdot (-5) \\ \widehat{LE} = & \big(\widehat\beta_0 - 5 \widehat\beta_2 \big)+ \\ & \big(\widehat\beta_1 - 5 \widehat\beta_3 \big) CP^c \end{aligned}\]

Vaccination rate of 91.45 %

\[\begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP^c + \\ & \widehat\beta_2 \cdot 0 + \\ & \widehat\beta_3 CP^c \cdot 0 \\ \widehat{LE} = & \big(\widehat\beta_0 \big)+ \\ & \big(\widehat\beta_1 \big) CP^c \end{aligned}\]

Vaccination rate of 96.45 %

\[\begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP^c + \\ & \widehat\beta_2 \cdot 5 + \\ & \widehat\beta_3 CP^c \cdot 5 \\ \widehat{LE} = & \big(\widehat\beta_0 + 5 \widehat\beta_2 \big)+ \\ & \big(\widehat\beta_1 + 5 \widehat\beta_3 \big) CP^c \end{aligned}\]

Poll Everywhere Question??

Interpretation for interaction between two continuous variables

\[ \begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP^c + \widehat\beta_2 VR^c + \widehat\beta_3 CP^c \cdot VR^c \\ \widehat{LE} = & \bigg[\widehat\beta_0 + \widehat\beta_2 \cdot VR^c \bigg] + \underbrace{\bigg[\widehat\beta_1 + \widehat\beta_3 \cdot VR^c \bigg]}_\text{CP's effect} CP \\ \end{aligned}\]

  • Interpretation:

    • \(\beta_3\) = mean change in cell phones’s effect, for every 1 % increase in vaccination rate

  • In summary, the interaction term can be interpreted as “difference in adjusted effect of number of cell phones for every 1 % increase in vaccination rate”

  • It will be helpful to test the interaction to round out this interpretation!!

Test interaction between two continuous variables (1/2)

  • We run an F-test for a single coefficients (\(\beta_3\)) in the below model (see lesson 9)

\[ LE = \beta_0 + \beta_1 CP^c + \beta_2 VR^c + \beta_3 CP^c \cdot VR^c + \epsilon\]

Null \(H_0\)

\[\beta_3=0\]

Alternative \(H_1\)

\[\beta_3\neq0\]

Null / Smaller / Reduced model

\[ LE = \beta_0 + \beta_1 CP^c + \beta_2 VR^c + \epsilon\]

Alternative / Larger / Full model

\[\begin{aligned} LE = & \beta_0 + \beta_1 CP^c + \beta_2 VR^c + \\ & \beta_3 CP^c \cdot VR^c + \epsilon \end{aligned}\]

Test interaction between two continuous variables (2/2)

  • Fit the reduced and full model
m_int_vr_red = gapm %>% 
  lm(formula = life_exp ~ CP_c + VR_c)
m_int_vr_full = gapm %>%
  lm(formula = life_exp ~ CP_c + VR_c + CP_c*VR_c)

 

Display the ANOVA table with F-statistic and p-value 
anova_table2 = anova(m_int_vr_red, m_int_vr_full) 
anova_table2 %>% tidy() %>% gt() %>% tab_options(table.font.size = 35) %>% 
  fmt_number(decimals = 3)
term df.residual rss df sumsq statistic p.value
life_exp ~ CP_c + VR_c 102.000 3,480.371 NA NA NA NA
life_exp ~ CP_c + VR_c + CP_c * VR_c 101.000 3,394.429 1.000 85.942 2.557 0.113
  • Conclusion: There is not a significant interaction between cell phones and vaccination rate (p = 0.113). Vaccination rate is not an effect modifier of the association between cell phones and life expectancy.

Learning Objective

Last time:

  1. Define confounders and effect modifiers, and how they interact with the main relationship we model.
  2. Interpret the interaction component of a model with a binary categorical covariate and continuous covariate, and how the main variable’s effect changes.
  3. Interpret the interaction component of a model with a multi-level categorical covariate and continuous covariate, and how the main variable’s effect changes.

This time:

  1. Interpret the interaction component of a model with two categorical covariates, and how the main variable’s effect changes.
  2. Interpret the interaction component of a model with two continuous covariates, and how the main variable’s effect changes.
  1. Report results for a best-fit line (with confidence intervals) at different levels of an effect measure modifier

How to find the confidence interval for each slope?

  • In the example with VR and CP, we showed:

Best-fit line for vaccination rate of 96.45 %

\[\begin{aligned} \widehat{LE} = & \big(\widehat\beta_0 + 5 \widehat\beta_2 \big)+ \big(\widehat\beta_1 + 5 \widehat\beta_3 \big) CP^c \end{aligned}\]

  • Often, we want to report the estimate of the combined coefficients: \(\widehat\beta_1 + 5 \widehat\beta_3\)

    • This allows us to make a statement like: “At a vaccination rate of 96.45%, mean life expectancy increases \(\big(\widehat\beta_1 + 5 \widehat\beta_3 \big)\) years for every one additional cell phone per 100 people (95% CI: __, __).”

 

  • We can calculate \(\widehat\beta_1 + 5 \widehat\beta_3\) by using the values of the estimated coefficients

  • BUT we always want to have a 95% confidence interval when we report this combined estimate!!

Getting a 95% confidence interval requires linear combinations!

  • If we want a confidence interval for \(\widehat\beta_1 + 5 \widehat\beta_3\), then we would use the formula:

\[\bigg(\widehat\beta_1 + 5 \widehat\beta_3 \bigg) \pm t^* \times SE_{(\beta_1 + 5 \beta_3)}\]

  • The hard part is figuring out what \(SE_{(\beta_1 + 5 \beta_3)}\) (or \(\text{Var}(\beta_1 + 5 \beta_3)\)) equals

  • We need to go back to variance of linear combinations (BSTA 511/611, EPI 525): \[\text{Var}(aX + bY) = a^2\text{Var}(X) + b^2\text{Var}(Y) + 2ab\text{Cov}(X, Y)\] or \[\text{Var}(aX - bY) = a^2\text{Var}(X) + b^2\text{Var}(Y) - 2ab\text{Cov}(X, Y)\]

Calculating \(SE_{(\beta_1 + 5 \beta_3)}\) “by hand” (REFERENCE)

  • A helpful function that returns the variance-covariance matric of all the coefficients in model m_int_vr:
vcov(m_int_vr)
              (Intercept)          CP_c          VR_c     CP_c:VR_c
(Intercept)  0.3842647156 -2.466140e-04 -1.098844e-02 -4.873521e-04
CP_c        -0.0002466140  3.357335e-04 -5.354918e-04  1.872476e-06
VR_c        -0.0109884449 -5.354918e-04  7.071783e-03  8.343243e-05
CP_c:VR_c   -0.0004873521  1.872476e-06  8.343243e-05  3.700339e-06

\[ \begin{aligned} \text{Var}(\beta_1) & = 3.35733\times 10^{-4} \\ \text{Var}(\beta_3) & = 0.0070718 \\ \text{Cov}(\beta_1, \beta_3) & = -5.35492\times 10^{-4} \\ \end{aligned}\]

\[ \begin{aligned} \text{Var}(\beta_1 + 5 \beta_3) & = \text{Var}(\beta_1) + 5 ^2 \text{Var}(\beta_3) + 2 \cdot 5 \cdot \text{Cov}(\beta_1, \beta_3) \\ \text{Var}(\beta_1 + 5 \beta_3) & = 3.35733\times 10^{-4} + 5^2 \ \times 3.7\times 10^{-6} + 2 \cdot 5 \cdot 1.872\times 10^{-6} \\ \text{Var}(\beta_1 + 5 \beta_3) & = 4.46967\times 10^{-4} \\ SE_{(\beta_1 + 5 \beta_3)} & = \sqrt{4.46967\times 10^{-4}} \\ SE_{(\beta_1 + 5 \beta_3)} & = 0.0211416 \end{aligned}\]

We can use R and estimable() to find the estimate and CI

For \(\widehat\beta_1 + 5 \widehat\beta_3\):

library(gmodels)
m_int_vr %>% estimable(
                   c("(Intercept)" = 0,      # beta0
                     "CP_c"       = 1,      # beta1
                     "VR_c"        = 0,      # beta2
                     "CP_c:VR_c"  = 5),  # beta3
                   conf.int = 0.95)
            Estimate Std. Error t value  DF     Pr(>|t|)   Lower.CI Upper.CI
(0 1 0 5) 0.09204476 0.02114159 4.35373 101 3.213735e-05 0.05010553 0.133984

 

Our conclusion: At a vaccination rate of 96.45 %, mean life expectancy increases 0.092 years for every one additional cell phone per 100 people (95% CI: 0.05, 0.134).

Another example: income (binary) and CP (1/2)

m_int_inc2 = gapm %>% 
  lm(formula = life_exp ~ CP_c*income_level_2)

\[ \begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP^c + \widehat\beta_2 I(\text{high income}) + \widehat\beta_3 CP^c \cdot I(\text{high income}) \\ \widehat{LE} = & 68.408 + 0.076 \cdot CP^c + 6.247 \cdot I(\text{high income}) -0.066 \cdot CP^c \cdot I(\text{high income}) \end{aligned}\]

For lower income countries: \(I(\text{high income}) =0\)

\[ \begin{aligned} \widehat{LE} = & \widehat\beta_0 + \widehat\beta_1 CP\\ \widehat{LE} = & 68.41 + 0.076 \cdot CP \end{aligned}\]

For higher income countries: \(I(\text{high income}) =1\)

\[ \begin{aligned} \widehat{LE} = & (\widehat\beta_0 +\widehat\beta_2) + (\widehat\beta_1 +\widehat\beta_3) CP \\ \widehat{LE} = & (68.41 + 6.247) + (0.076 -0.066) \cdot CP \\ \widehat{LE} = & 74.65 + 0.01 \cdot CP \end{aligned}\]

Another example: income (binary) and CP (2/2)

names(m_int_inc2$coefficients) # I just need to see the exact names
[1] "(Intercept)"                      "CP_c"                            
[3] "income_level_2Higher income"      "CP_c:income_level_2Higher income"
m_int_inc2 %>% estimable(
                   c("(Intercept)" = 0,      # beta0
                     "CP_c"       = 1,      # beta1
                     "income_level_2Higher income" = 0,      # beta2
                     "CP_c:income_level_2Higher income"  = 1),  # beta3
                   conf.int = 0.95)
            Estimate Std. Error   t value  DF  Pr(>|t|)    Lower.CI   Upper.CI
(0 1 0 1) 0.01001064 0.02723878 0.3675143 101 0.7140044 -0.04402377 0.06404504

 

Our conclusion: For countries with high income, mean life expectancy increases 0.01 years for every one additional cell phone per 100 people (95% CI: -0.044, 0.064).

If our example had an effect measure modifier

  • None of our examples had a significant interaction, so it’s hard to demonstrate exactly how we would report this

  • Let’s say, just for example, that income had a significant interaction with CP

    • How would we report this to an audience??

  • Here’s how to report on an interaction/EMM:

    • We found that a country’s income status (high or low) is a significant effect measure modifier on number of cell phones (include p-value for interaction test here). For countries with high income, mean life expectancy increases 0.01 years for every additional cell phone per 100 people (95% CI: -0.044, 0.064). For countries with low income, mean life expectancy increases 6.247 years for every additional cell phone per 100 people (95% CI: 3.825, 8.668).