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")))
Table 3.1
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.")
Table 3.2
Summary Statistics

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. N = 10. NA = not available or missing; Mdn = median; Q1 = 25th percentile; Q3 = 75th percentile. 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.

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)
Table 3.3
Correlations

Variable Pair

r

p

Exercise Frequency

Weight Loss

.860

.001**

Food Intake

Weight Loss

.047

.898

Food Intake

Exercise Frequency

.380

.282

Note. N = 10. r = Pearson's Product-Moment correlation coefficient.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.

* 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

summary(fit_lm_exer)

Call:
lm(formula = loss ~ exer, data = df_loss)

Residuals:
   Min     1Q Median     3Q    Max 
 -2.50  -1.50   0.25   1.25   2.00 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   4.0000     0.9129   4.382  0.00234 **
exer          1.7500     0.3608   4.850  0.00127 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.768 on 8 degrees of freedom
Multiple R-squared:  0.7462,    Adjusted R-squared:  0.7145 
F-statistic: 23.52 on 1 and 8 DF,  p-value: 0.001272
summary(fit_lm_diet)

Call:
lm(formula = loss ~ diet, data = df_loss)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.4286 -2.3571  0.5714  1.5000  5.4286 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept)  7.14286    2.92258   2.444   0.0403 *
diet         0.07143    0.54085   0.132   0.8982  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.505 on 8 degrees of freedom
Multiple R-squared:  0.002175,  Adjusted R-squared:  -0.1226 
F-statistic: 0.01744 on 1 and 8 DF,  p-value: 0.8982
summary(fit_lm_both)

Call:
lm(formula = loss ~ exer + diet, data = df_loss)

Residuals:
   Min     1Q Median     3Q    Max 
    -2     -1      0      1      2 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   6.0000     1.2749   4.706 0.002193 ** 
exer          2.0000     0.3333   6.000 0.000542 ***
diet         -0.5000     0.2520  -1.984 0.087623 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.512 on 7 degrees of freedom
Multiple R-squared:  0.8376,    Adjusted R-squared:  0.7912 
F-statistic: 18.05 on 2 and 7 DF,  p-value: 0.001727
summary(fit_lm_three)

Call:
lm(formula = loss ~ exer + diet + meta, data = df_loss)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.0000 -0.6136  0.0000  0.3409  2.0000 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  -1.6364     2.9285  -0.559  0.59654   
exer          1.0455     0.4223   2.476  0.04808 * 
diet         -1.1364     0.2942  -3.863  0.00834 **
meta          0.6364     0.2318   2.746  0.03349 * 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.087 on 6 degrees of freedom
Multiple R-squared:  0.928, Adjusted R-squared:  0.892 
F-statistic: 25.78 on 3 and 6 DF,  p-value: 0.0007937

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("Exercise"    = fit_lm_exer, 
                                 "Food Intake" = fit_lm_diet, 
                                 "Both"        = fit_lm_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
Table 3.4
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

AIC

43.5

57.2

41.1

BIC

44.4

58.1

42.3

R²

.746

.002

.838

Adjusted R²

.714

< .001

.791

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("Two Predictors" = fit_lm_both, 
                      "Three Predictors" = fit_lm_three),
                 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. ")
Table 3.5
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*

AIC

41.1

34.9

BIC

42.3

36.5

R²

.838

.928

Adjusted R²

.791

.892

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

3.4.4 Cohen’s \(f\) (see chapter 8)

Cohen’s \(f\) (Cohen, 1988) is appropriate for calculating the effect size within a multiple regression model in which the independent variable of interest and the dependent variable are both continuous.

effectsize::cohens_f_squared(fit_lm_both,
                             partial = TRUE)
# A tibble: 2 × 5
  Parameter Cohens_f2_partial    CI CI_low CI_high
  <chr>                 <dbl> <dbl>  <dbl>   <dbl>
1 exer                  4.59   0.95  0.976     Inf
2 diet                  0.563  0.95  0         Inf
effectsize::cohens_f_squared(fit_lm_both,
                             partial = FALSE)
# A tibble: 2 × 5
  Parameter Cohens_f2    CI CI_low CI_high
  <chr>         <dbl> <dbl>  <dbl>   <dbl>
1 exer          2.94   0.95  0.506     Inf
2 diet          0.101  0.95  0         Inf

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

apaSupp::tab_lm(fit_lm_both, vif = TRUE)
Table 3.6
Parameter Estimates for Linear Regression

b

(SE)

p

b*

VIF

η²

ηₚ²

(Intercept)

6.00

(1.27)

.002**

exer

2.00

(0.33)

< .001***

0.99

1.17

.835

.837

diet

-0.50

(0.25)

.088

-0.33

1.17

.091

.360

R²

.838

Adjusted R²

.791

Note. N = 10. VIF = variance inflation factor; η² = semi-partial correlation; ηₚ² = partial correlation; b* = standardize coefficient; p = significance from Wald t-test for parameter estimate.

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

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 1.350  7     6.81    13.19
    1    9.5 1.120  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].