3 D&H Ch3a - Multiple Regression: “Weight”

Darlington & Hayes, Chapter 3’s first example

Darlington, Richard B., and Andrew F. Hayes. Regression Analysis and Linear Models : Concepts, Applications, and Implementation, Guilford Publications, 2016. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/usu/detail.action?docID=4652287. Created from usu on 2025-01-29 01:13:21.

Outside resource * https://www.harshaash.com/R/part-and-partial-corr/ * https://dstanley4.github.io/psyc4780bookdown/multiple-regression.html

Reference: 

# install.packages("remotes")
# remotes::install_github("sarbearschwartz/apaSupp")
# remotes::install_github("ddsjoberg/gtsummary")

library(magrittr)       
library(tidyverse)   
library(broom)     
library(naniar)
library(corrplot)   
library(GGally)
library(gtsummary)
library(flextable)
library(apaSupp)
library(performance)
library(interactions)
library(effects)
library(emmeans)
library(car)
library(ggResidpanel)
library(modelsummary)
library(ppcor)
library(jtools)
library(olsrr)
library(DescTools)
library(effectsize)
library(ggpubr)

3.1 PURPOSE

3.1.1 Research Question

How is weight loss associationed with exercise frequency and food intake?

3.1.2 Data Description

Suppose you conducted a study examining the relationship between food consumption and weight loss among people enrolled (n = 10) in a month-long healthy living class.

Dependent Variable (DV)

  • loss average weight loss in hundreds of grams per week

Independent Variables (IVs)

  • exer average weekly hours of exercise

  • diet average daily food consumption (in 100s of calories about the recommended minimum of 1,000 calories required to maintain good health)

  • meta metabolic rate

Manually enter the data set provided on page 44 in Table 3.1 

df_loss <- tibble::tribble(~id, ~exer, ~diet, ~meta, ~loss,
                            1, 0, 2, 15,  6, 
                            2, 0, 4, 14,  2,
                            3, 0, 6, 19,  4,
                            4, 2, 2, 15,  8,
                            5, 2, 4, 21,  9,
                            6, 2, 6, 23,  8,
                            7, 2, 8, 21,  5,
                            8, 4, 4, 22, 11,
                            9, 4, 6, 24, 13,
                           10, 4, 8, 26,  9)
df_loss %>% 
  dplyr::select("ID" = id,
                "Exercise\nFrequency" = exer,
                "Food\nIntake" = diet,
                "Weight\nLoss" = loss) %>% 
  flextable::flextable() %>% 
  apaSupp::theme_apa(caption = "D&H Table 3.1: Exercise, Food Intake, and Weight Loss") %>% 
  flextable::colformat_double(digits = 0) %>% 
  flextable::footnote(part = "header", 
                      i = 1, j = 2:4, 
                      value = flextable::as_paragraph(c("Weekly average of hours/day",
                                             "Daily average of 100s of calories above recommendation",
                                             "Weekly average of 100s of grams/week")))
``` ```{=html}
(\#tab:unnamed-chunk-116)D&H Table 3.1: Exercise, Food Intake, and Weight Loss

ID

Exercise
Frequency1

Food
Intake2

Weight
Loss3

1

0

2

6

2

0

4

2

3

0

6

4

4

2

2

8

5

2

4

9

6

2

6

8

7

2

8

5

8

4

4

11

9

4

6

13

10

4

8

9

1Weekly average of hours/day

2Daily average of 100s of calories above recommendation

3Weekly average of 100s of grams/week

3.2 EXPLORATORY DATA ANALYSIS

3.2.1 Summary Statistics

3.2.1.1 Univariate

df_loss %>% 
  dplyr::select("Exercise Frequency" = exer,
                "Food Intake" = diet,
                "Metabolic Rate" = meta,
                "Weight Loss" = loss) %>% 
  apaSupp::tab_desc(caption = "Summary Statistics",
                    general_not = "Exercise captures daily average hours.  Food intake is the average of 100's of calories above the recommendation.  Metabolism is the metobalic rate. Weight loss is average weekly weight lost in 100s of grams.")
``` ```{=html}
(\#tab:unnamed-chunk-117)Summary Statistics

Measure

NA

M

SD

min

Q1

Mdn

Q3

max

Exercise Frequency

0

2.00

1.63

0.00

0.50

2.00

3.50

4.00

Food Intake

0

5.00

2.16

2.00

4.00

5.00

6.00

8.00

Metabolic Rate

0

20.00

4.14

14.00

16.00

21.00

22.75

26.00

Weight Loss

0

7.50

3.31

2.00

5.25

8.00

9.00

13.00

Note. Exercise captures daily average hours. Food intake is the average of 100's of calories above the recommendation. Metabolism is the metobalic rate. Weight loss is average weekly weight lost in 100s of grams. NA = not available or missing. Mdn = median. Q1 = 25th percentile, Q3 = 75th percentile. N = 10.

3.2.1.2 Bivariate

df_loss %>% 
  dplyr::select("Weight Loss" = loss,
                "Exercise Frequency" = exer,
                "Food Intake" = diet) %>% 
  apaSupp::tab_cor(caption = "Correlations",
                   general_note = "Exercise captures daily average hours.  Food intake is the average of 100's of calories above the recommendation.  Weight loss is average weekly weight lost in 100s of grams.") %>% 
  flextable::hline(i = 2)
``` ```{=html}
(\#tab:unnamed-chunk-118)Correlations

Variables

r

p

Weight Loss

Exercise Frequency

0.860

.001

**

Weight Loss

Food Intake

0.047

.898

Exercise Frequency

Food Intake

0.380

.282

Note. Exercise captures daily average hours. Food intake is the average of 100's of calories above the recommendation. Weight loss is average weekly weight lost in 100s of grams. r = Pearson's Product-Moment correlation coefficient. N = 10.

* p < .05. ** p < .01. *** p < .001.

3.2.2 Visualizations

3.2.2.1 Univariate

df_loss %>% 
  dplyr::select(id, 
                "Weight Loss\n(100 g/day)" = loss,
                "Exercise Frequency\n(hr/day)" = exer,
                "Food Intake\n(100 cal/day above 1000)" = diet,
                "Metabolic Rate" = meta) %>% 
  tidyr::pivot_longer(cols = -id) %>% 
  ggplot(aes(value)) +
  geom_histogram(binwidth = 1,
                 color = "black",
                 alpha = .25) +
  theme_bw() +
  facet_wrap(~ name,
             scale = "free_x") +
  labs(x = NULL,
       y = "Count")

Figure 3.1
Univariate Distibution on Continuous Measures

Univariate Distibution on Continuous Measures

3.2.2.2 Bivariate

df_loss %>% 
  ggplot(aes(x = diet,
             y = loss)) +
  geom_point(aes(shape = factor(exer)))

Figure 3.2
D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - BASIC

D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - BASIC 
df_loss %>% 
  ggplot(aes(x = diet,
             y = loss)) +
  geom_point(aes(shape = factor(exer),
                 color = factor(exer)),
             size = 3)  +
  theme_bw() +
  labs(x = "Observed Food Intake)",
       y = "Observed Weight Loss",
       shape = "Exercise Frequency",
       color = "Exercise Frequency") +
  theme(legend.position = "bottom")

Figure 3.3
D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - BETTTER

D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - BETTTER 
df_loss %>% 
  dplyr::mutate(exer = factor(exer)) %>% 
  ggplot(aes(x = diet,
             y = loss)) +
  geom_point(size = 3)  +
  theme_bw() +
  labs(x = "Observed Food Intake)",
       y = "Observed Weight Loss",
       shape = "Exercise Frequency",
       color = "Exercise Frequency") +
  theme(legend.position = "bottom") +
  geom_smooth(aes(group = 1),
              method = "lm",
              formula = y ~ x,
              se = FALSE)

Figure 3.4
D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - WORST

D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - WORST 
df_loss %>% 
  dplyr::mutate(exer = factor(exer)) %>% 
  ggplot(aes(x = diet,
             y = loss,
             group = exer)) +
  geom_point(aes(shape = exer,
                 color = exer),
             size = 3)  +
  theme_bw() +
  labs(x = "Observed Food Intake",
       y = "Observed Weight Loss",
       shape = "Exercise Frequency",
       color = "Exercise Frequency") +
  theme(legend.position = "bottom") +
  geom_smooth(aes(color = exer),
              method = "lm",
              formula = y ~ x,
              se = FALSE)

Figure 3.5
D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - BEST

D&H Figure 3.1 (page 45) An example with positive simple association and negative partial association - BEST

3.3 REGRESSION ANALYSIS

3.3.1 Fit the models

  • The dependent variable (DV) is weight loss (\(Y\))
  • The independent variables (IVs) are exercise and diet (\(X\))
fit_lm_exer <- lm(loss ~ exer,
                  data = df_loss)

fit_lm_diet <- lm(loss ~ diet,
                  data = df_loss)

fit_lm_both <- lm(loss ~ exer + diet,
                  data = df_loss)

fit_lm_three<- lm(loss ~ exer + diet + meta,
                  data = df_loss)

3.3.2 Output

olsrr::ols_regress(fit_lm_exer)
                        Model Summary                          
--------------------------------------------------------------
R                       0.864       RMSE                1.581 
R-Squared               0.746       MSE                 2.500 
Adj. R-Squared          0.714       Coef. Var          23.570 
Pred R-Squared          0.585       AIC                43.542 
MAE                     1.300       SBC                44.449 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                              ANOVA                                
------------------------------------------------------------------
               Sum of                                             
              Squares       DF    Mean Square      F         Sig. 
------------------------------------------------------------------
Regression     73.500        1         73.500     23.52    0.0013 
Residual       25.000        8          3.125                     
Total          98.500        9                                    
------------------------------------------------------------------

                                Parameter Estimates                                  
------------------------------------------------------------------------------------
      model     Beta    Std. Error    Std. Beta      t       Sig     lower    upper 
------------------------------------------------------------------------------------
(Intercept)    4.000         0.913                 4.382    0.002    1.895    6.105 
       exer    1.750         0.361        0.864    4.850    0.001    0.918    2.582 
------------------------------------------------------------------------------------
olsrr::ols_regress(fit_lm_diet)
                         Model Summary                          
---------------------------------------------------------------
R                        0.047       RMSE                3.135 
R-Squared                0.002       MSE                 9.829 
Adj. R-Squared          -0.123       Coef. Var          46.735 
Pred R-Squared          -0.393       AIC                57.232 
MAE                      2.600       SBC                58.139 
---------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                              ANOVA                               
-----------------------------------------------------------------
               Sum of                                            
              Squares       DF    Mean Square      F        Sig. 
-----------------------------------------------------------------
Regression      0.214        1          0.214    0.017    0.8982 
Residual       98.286        8         12.286                    
Total          98.500        9                                   
-----------------------------------------------------------------

                                 Parameter Estimates                                   
--------------------------------------------------------------------------------------
      model     Beta    Std. Error    Std. Beta      t       Sig      lower     upper 
--------------------------------------------------------------------------------------
(Intercept)    7.143         2.923                 2.444    0.040     0.403    13.882 
       diet    0.071         0.541        0.047    0.132    0.898    -1.176     1.319 
--------------------------------------------------------------------------------------
olsrr::ols_regress(fit_lm_both)
                        Model Summary                          
--------------------------------------------------------------
R                       0.915       RMSE                1.265 
R-Squared               0.838       MSE                 1.600 
Adj. R-Squared          0.791       Coef. Var          20.158 
Pred R-Squared          0.669       AIC                41.079 
MAE                     1.200       SBC                42.289 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                              ANOVA                                
------------------------------------------------------------------
               Sum of                                             
              Squares       DF    Mean Square      F         Sig. 
------------------------------------------------------------------
Regression     82.500        2         41.250    18.047    0.0017 
Residual       16.000        7          2.286                     
Total          98.500        9                                    
------------------------------------------------------------------

                                  Parameter Estimates                                   
---------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower    upper 
---------------------------------------------------------------------------------------
(Intercept)     6.000         1.275                  4.706    0.002     2.985    9.015 
       exer     2.000         0.333        0.987     6.000    0.001     1.212    2.788 
       diet    -0.500         0.252       -0.326    -1.984    0.088    -1.096    0.096 
---------------------------------------------------------------------------------------
olsrr::ols_regress(fit_lm_three)
                        Model Summary                          
--------------------------------------------------------------
R                       0.963       RMSE                0.842 
R-Squared               0.928       MSE                 0.709 
Adj. R-Squared          0.892       Coef. Var          14.495 
Pred R-Squared          0.836       AIC                34.941 
MAE                     0.655       SBC                36.454 
--------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                              ANOVA                                
------------------------------------------------------------------
               Sum of                                             
              Squares       DF    Mean Square      F         Sig. 
------------------------------------------------------------------
Regression     91.409        3         30.470    25.782     8e-04 
Residual        7.091        6          1.182                     
Total          98.500        9                                    
------------------------------------------------------------------

                                  Parameter Estimates                                    
----------------------------------------------------------------------------------------
      model      Beta    Std. Error    Std. Beta      t        Sig      lower     upper 
----------------------------------------------------------------------------------------
(Intercept)    -1.636         2.928                 -0.559    0.597    -8.802     5.529 
       exer     1.045         0.422        0.516     2.476    0.048     0.012     2.079 
       diet    -1.136         0.294       -0.742    -3.863    0.008    -1.856    -0.417 
       meta     0.636         0.232        0.796     2.746    0.033     0.069     1.203 
----------------------------------------------------------------------------------------

3.3.3 Equations

\[ \hat{loss} = 4 + 1.75(exercise) \\ \hat{loss} = 7.14 + 0.07(food) \\ \hat{loss} = 6 + 2(exercise) -0.5(food) \\ \hat{loss} = -1.636 + 1.045(exercise) -1.136(food) + 0.636(metabolism)\\ \]

3.3.4 Tables

tab_lm3 <- apaSupp::tab_lms(list(fit_lm_exer, 
                                 fit_lm_diet, 
                                 fit_lm_both),
                            mod_names = c("Exercise", 
                                          "Food Intake", 
                                          "Both"),
                            var_labels = c("exer" = "Exercise",
                                           "diet" = "Food"),
                            caption = "Parameter Estimates for Weight Loss Regressed on Exercise Frequency and Food Intake",
                            general_note = "Dependent variable is average weekly weight lost in 100s of grams. Exercise is daily average hours and food intake is the average of 100's of calories above the recommendation. ") 


tab_lm3
``` ```{=html}
(\#tab:unnamed-chunk-129)Parameter Estimates for Weight Loss Regressed on Exercise Frequency and Food Intake

Exercise

Food Intake

Both

Variable

b

(SE)

p

b

(SE)

p

b

(SE)

p

(Intercept)

4.00

(0.91)

.002 **

7.14

(2.92)

.040 *

6.00

(1.27)

.002 **

Exercise

1.75

(0.36)

.001 **

2.00

(0.33)

< .001 ***

Food

0.07

(0.54)

.898

-0.50

(0.25)

.088

R²

0.75

0.00

0.84

Adjusted R²

0.71

-0.12

0.79

Note. Dependent variable is average weekly weight lost in 100s of grams. Exercise is daily average hours and food intake is the average of 100's of calories above the recommendation.

* p < .05. ** p < .01. *** p < .001.

How to save to word document in the same folder as your R file.

flextable::save_as_docx(tab_lm3,
                        path = "DH_ch3_tab_lm3.docx")
apaSupp::tab_lms(list(fit_lm_both, fit_lm_three),
                 mod_names = c("Two Predictors", "Three Predictors"),
                 var_labels = c("exer" = "Exercise Freq",
                                "diet" = "Food Intake",
                                "meta" = "Metabolic Rate"),
                 caption = "Parameter Estimates for Weight Loss Regressed on Exercise Frequency, Food Intake, and Metabolic Rate",
                            general_note = "Dependent variable is average weekly weight lost in 100s of grams. Exercise is daily average hours and food intake is the average of 100's of calories above the recommendation. ")
``` ```{=html}
(\#tab:unnamed-chunk-131)Parameter Estimates for Weight Loss Regressed on Exercise Frequency, Food Intake, and Metabolic Rate

Two Predictors

Three Predictors

Variable

b

(SE)

p

b

(SE)

p

(Intercept)

6.00

(1.27)

.002 **

-1.64

(2.93)

.597

Exercise Freq

2.00

(0.33)

< .001 ***

1.05

(0.42)

.048 *

Food Intake

-0.50

(0.25)

.088

-1.14

(0.29)

.008 **

Metabolic Rate

0.64

(0.23)

.033 *

R²

0.84

0.93

Adjusted R²

0.79

0.89

Note. Dependent variable is average weekly weight lost in 100s of grams. Exercise is daily average hours and food intake is the average of 100's of calories above the recommendation.

* p < .05. ** p < .01. *** p < .001.

3.3.5 Visualization

Figure 3.6
Darlington & Hayes: Figure 3.2, page 49

Darlington & Hayes: Figure 3.2, page 49 
interactions::interact_plot(model = fit_lm_both,
                            pred = diet,
                            modx = exer,
                            modx.values = 0:4) +
  theme_bw() 

interactions::interact_plot(model = fit_lm_both,
                            pred = exer,
                            modx = diet,
                            modx.values = c(0, 2, 4, 6, 8)) +
  theme_bw() 

interactions::interact_plot(model = fit_lm_three,
                            pred = exer,
                            modx = diet,
                            mod2 = meta,
                            modx.values = c(8, 5, 2),
                            modx.labels = c("1800 cal",
                                            "1500 cal",
                                            "1200 cal"),
                            mod2.values = c(16, 22.75),
                            mod2.labels = c("Low Metabolic Rate\n(Q1 = 16.00)", 
                                            "High Metabolic Rate\n(Q3 = 22.75)"),
                            x.label = "Daily Exercise, hours",
                            y.label = "Weight Loss, 100s grams/week",
                            legend.main = "Food Consumption") +
  theme_bw() +
  theme(legend.position = c(0, 1),
        legend.justification = c(-0.1, 1.1),
        legend.background = element_rect(color = "black"),
        legend.key.width = unit(2, "cm")) +
  scale_y_continuous(breaks = c(0, 5, 10, 15),
                     labels = c(0, 500, 1000, 1500))

3.4 EFFECT SIZES

3.4.1 Semipartial Correlation

Unique contribution

effectsize::r2_semipartial(fit_lm_both)
# A tibble: 2 × 5
  Term  r2_semipartial    CI CI_low CI_high
  <chr>          <dbl> <dbl>  <dbl>   <dbl>
1 exer          0.835   0.95  0.675       1
2 diet          0.0914  0.95  0           1
effectsize::r2_semipartial(fit_lm_three)
# A tibble: 3 × 5
  Term  r2_semipartial    CI   CI_low CI_high
  <chr>          <dbl> <dbl>    <dbl>   <dbl>
1 exer          0.0735  0.95 0.000735       1
2 diet          0.179   0.95 0.00179        1
3 meta          0.0904  0.95 0.000904       1

3.4.2 Eta Squared

effectsize::eta_squared(fit_lm_both, partial = FALSE)
# A tibble: 2 × 5
  Parameter   Eta2    CI CI_low CI_high
  <chr>      <dbl> <dbl>  <dbl>   <dbl>
1 exer      0.746   0.95  0.336       1
2 diet      0.0914  0.95  0           1
effectsize::eta_squared(fit_lm_three, partial = FALSE)
# A tibble: 3 × 5
  Parameter   Eta2    CI CI_low CI_high
  <chr>      <dbl> <dbl>  <dbl>   <dbl>
1 exer      0.746   0.95  0.284       1
2 diet      0.0914  0.95  0           1
3 meta      0.0904  0.95  0           1
effectsize::eta_squared(fit_lm_both, partial = TRUE)
# A tibble: 2 × 5
  Parameter Eta2_partial    CI CI_low CI_high
  <chr>            <dbl> <dbl>  <dbl>   <dbl>
1 exer             0.821  0.95  0.494       1
2 diet             0.36   0.95  0           1
effectsize::eta_squared(fit_lm_three, partial = TRUE)
# A tibble: 3 × 5
  Parameter Eta2_partial    CI CI_low CI_high
  <chr>            <dbl> <dbl>  <dbl>   <dbl>
1 exer             0.912  0.95 0.699        1
2 diet             0.559  0.95 0.0445       1
3 meta             0.557  0.95 0.0425       1

3.4.3 Standardized Regression Coefficients

parameters::standardise_parameters(fit_lm_both)
# A tibble: 3 × 5
  Parameter   Std_Coefficient    CI CI_low CI_high
  <chr>                 <dbl> <dbl>  <dbl>   <dbl>
1 (Intercept)        8.24e-17  0.95 -0.342  0.342 
2 exer               9.87e- 1  0.95  0.598  1.38  
3 diet              -3.26e- 1  0.95 -0.716  0.0626
parameters::standardise_parameters(fit_lm_three)
# A tibble: 4 × 5
  Parameter   Std_Coefficient    CI   CI_low CI_high
  <chr>                 <dbl> <dbl>    <dbl>   <dbl>
1 (Intercept)        1.65e-16  0.95 -0.254     0.254
2 exer               5.16e- 1  0.95  0.00601   1.03 
3 diet              -7.42e- 1  0.95 -1.21     -0.272
4 meta               7.96e- 1  0.95  0.0866    1.50

3.5 VARIANCE INFLATION FACTORS

car::vif(fit_lm_both)
    exer     diet 
1.166667 1.166667 
car::vif(fit_lm_three)
    exer     diet     meta 
3.621212 3.075758 7.000000

3.6 CONCLUSION: SPECIFIC QUESTIONS

coef(fit_lm_both)
(Intercept)        exer        diet 
        6.0         2.0        -0.5 
confint(fit_lm_both)
                2.5 %     97.5 %
(Intercept)  2.985316 9.01468433
exer         1.211792 2.78820808
diet        -1.095829 0.09582931

\[ \hat{loss} = 6 + 2(exercise) -0.5(food) \]

Suppose an examination of the literature on diet, exercise, and weight loss led you to the following conclusions:

  1. People who eat the minimum number of calories to maintain good health and who do not exercise at all will lose how much?
fit_lm_both %>% 
  emmeans::emmeans(~ exer + diet,
                   at = list(exer = 0,
                             diet = 0))
 exer diet emmean   SE df lower.CL upper.CL
    0    0      6 1.27  7     2.99     9.01

Confidence level used: 0.95

The estimated marginal mean weight loss is 600 grams per week among thoes who eat the minimum number of calories to maintain good health and who do not exercise at, b = 6.00, 95% CI = [2.99, 9.01].

  1. If food intake is held constant, then each 1 hour of average daily exercise leads to how much more weight loss?
fit_lm_both %>% 
  emmeans::emmeans(~ exer + diet,
                   at = list(exer = 0:4))
 exer diet emmean    SE df lower.CL upper.CL
    0    5    3.5 0.820  7     1.56     5.44
    1    5    5.5 0.583  7     4.12     6.88
    2    5    7.5 0.478  7     6.37     8.63
    3    5    9.5 0.583  7     8.12    10.88
    4    5   11.5 0.820  7     9.56    13.44

Confidence level used: 0.95

Holding food consumption constant, each additional hour of daily exercise is associated with 200 grams additional weight lost per week, b = 2.00, 95% CI = [1.21, 2.79].

  1. If exercise is held constant, then each one unit of food intake per day (i.e., 100 calories) above the minimum to maintain good health translates to much weight gain?
fit_lm_both %>% 
  emmeans::emmeans(~ diet,
                   at = list(diet = 0:4))
 diet emmean    SE df lower.CL upper.CL
    0   10.0 1.348  7     6.81    13.19
    1    9.5 1.116  7     6.86    12.14
    2    9.0 0.894  7     6.89    11.11
    3    8.5 0.695  7     6.86    10.14
    4    8.0 0.540  7     6.72     9.28

Confidence level used: 0.95

Holding exercise constant, each additional 100 calories of food consumed daily is associated with 50 grams weight gained per week, b = -0.50, 95% CI = [1.21, 2.79].

  1. If a person exercises 2 hours per week and eats 600 more calories than the minimum, how much weight would they loose weekly on average?
fit_lm_both %>% 
  emmeans::emmeans(~ exer + diet,
                   at = list(exer = 2,
                             diet = 6))
 exer diet emmean   SE df lower.CL upper.CL
    2    6      7 0.54  7     5.72     8.28

Confidence level used: 0.95

On average, 700 grams were lost per week amonth participants who exrcised 2 hours a week and consumed 6,600 more calories, 95% CI [572, 828].