1 Scenario and Dataset

A school psychologist wants to know whether recreational screen time relates to anxiety symptoms in adolescents once physical activity and parental monitoring are taken into account, and whether monitoring changes how strongly screen time relates to anxiety.

You have anonymized records for 140 adolescents (13-16 years): weekly screen time, weekly moderate-to-vigorous physical activity, parental monitoring (“High”/“Low”), and an anxiety-symptom score. Set up and interpret the analysis independently, following the same logic as the lab practical – as a model for how to reason through each step, not a template to copy mechanically, since the variables and one of the conclusions differ here.

data/screen_time_anxiety_homework.csv contains 140 rows, one per adolescent.

Variable Type Description
adolescent_id Integer Anonymous participant identifier, 1-140
screen_time Continuous Average daily recreational screen time, in hours
physical_activity Continuous Hours per week spent in moderate-to-vigorous physical activity
monitoring Categorical (2 levels) Parent-reported level of monitoring of the adolescent’s screen use: “High” or “Low”
anxiety_score Continuous Score on a standardized adolescent anxiety-symptom scale, 0-100

2 Questions

These questions ask for more independent judgment than the lab practical. In several places you are asked to choose or justify a step yourself, rather than being told which one to take. Use data/screen_time_anxiety_homework.csv.

  1. Report the sample size and, separately for each monitoring group, the mean and standard deviation of screen_time, physical_activity, and anxiety_score. Also report the correlation between screen_time and physical_activity, and explain what it implies for interpreting a model that includes both.
  2. Does daily screen time predict anxiety symptoms? Fit the appropriate simple regression model. Report and interpret the slope, with correct units, and state whether the effect is statistically significant.
  3. Does physical activity explain anxiety symptoms beyond screen time? Extend the model from Question 2. Report the model, test whether the extension significantly improves it (show your work, as in the lab practical, not just the final \(F\) and \(p\)), and interpret the sign of the physical-activity coefficient in substantive terms.
  4. Is parental monitoring associated with anxiety, controlling for screen time and physical activity? Add monitoring to the model as a dummy predictor. State explicitly which level you coded as the reference category and why that choice does or does not matter for the conclusions you can draw. Report the coefficient, its test, and the model improvement over Question 3’s model.
  5. Compute standardized coefficients for the two continuous predictors in the Question 4 model. Which has the larger standardized association with anxiety score?
  6. Check the LINE assumptions for the model from Question 4, using the same set of diagnostic plots and statistics used in the lab practical. Report your verdict on each of the four assumptions, with the specific statistic or plot feature that supports each verdict.
  7. Identify the observation with the largest absolute residual in the Question 4 model. Is it unduly influential? Justify your answer with an appropriate statistic, not the residual alone.
  8. Does the association between screen time and anxiety differ by monitoring level? Extend the model from Question 4 with the relevant interaction term. Report the coefficients and test whether the interaction significantly improves the model.
  9. This is the point at which this homework’s dataset behaves differently from the lab practical’s. Depending on what you found in Question 8, do one of the following:
    • If the interaction is statistically significant, compute and interpret the simple slopes of screen time on anxiety, separately for each monitoring level, as in the lab practical.
    • If the interaction is not statistically significant, explain what the correct conclusion is. Is it appropriate to still describe, in the results write-up, how the two simple slopes “look” different from one another? Justify your answer with reference to what a non-significant interaction test actually tells you.
  10. Write a short results paragraph, addressed to the school psychologist, summarizing what the full set of models supports and does not support, including an explicit statement about whether these data can establish that screen time causes anxiety symptoms.

3 Guided Walkthrough

3.1 Inspecting the data

d <- read.csv("data/screen_time_anxiety_homework.csv", stringsAsFactors = TRUE)
d$monitoring <- relevel(d$monitoring, ref = "Low")
str(d)
## 'data.frame':    140 obs. of  5 variables:
##  $ adolescent_id    : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ screen_time      : num  4.6 3.5 2.3 6.6 5.9 3.8 7 6.6 3.6 4.2 ...
##  $ physical_activity: num  5.6 6.8 4.4 4.5 4.5 4.2 8.8 3.3 6.2 7.8 ...
##  $ monitoring       : Factor w/ 2 levels "Low","High": 2 2 2 2 2 2 2 1 1 1 ...
##  $ anxiety_score    : num  9.5 26.2 30.6 26 20.3 13.4 23.6 44.4 35.3 48.6 ...
d %>%
  group_by(monitoring) %>%
  summarise(n = n(),
            screen_time_M = mean(screen_time), screen_time_SD = sd(screen_time),
            physical_activity_M = mean(physical_activity), physical_activity_SD = sd(physical_activity),
            anxiety_score_M = mean(anxiety_score), anxiety_score_SD = sd(anxiety_score)) %>%
  kable()
monitoring n screen_time_M screen_time_SD physical_activity_M physical_activity_SD anxiety_score_M anxiety_score_SD
Low 75 4.368 1.618 5.023 2.649 40.25 11.22
High 65 4.705 1.537 4.877 2.363 23.03 10.48 
cor(d$screen_time, d$physical_activity)
## [1] -0.2336

The negative correlation between screen time and physical activity is modest, consistent with adolescents who spend more time on screens also spending somewhat less time active. It is small enough that entering both predictors together should not create serious collinearity, but it is not negligible, so each coefficient below should still be read as “controlling for the other,” not as if the two predictors were unrelated.

ggplot(d, aes(screen_time, anxiety_score, color = monitoring)) +
  geom_point(alpha = 0.75) +
  scale_color_manual(values = c("Low" = "grey40", "High" = ntnublue)) +
  labs(x = "Daily screen time (hours)", y = "Anxiety score",
       title = "Screen time and anxiety score, by parental monitoring level", color = "Monitoring")

3.2 Model 1 – does daily screen time predict anxiety symptoms?

m1 <- lm(anxiety_score ~ screen_time, data = d)
summary(m1)
## 
## Call:
## lm(formula = anxiety_score ~ screen_time, data = d)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -30.972  -9.095   0.285   9.583  30.932 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   20.265      3.398    5.96    2e-08 ***
## screen_time    2.649      0.709    3.74  0.00027 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 13.2 on 138 degrees of freedom
## Multiple R-squared:  0.0918, Adjusted R-squared:  0.0853 
## F-statistic:   14 on 1 and 138 DF,  p-value: 0.000273
confint(m1)
##              2.5 % 97.5 %
## (Intercept) 13.545 26.984
## screen_time  1.247  4.052

Each additional hour of daily screen time is associated with a 2.65-point increase in predicted anxiety score, on average. The effect is statistically significant, though screen time alone accounts for a modest 9.2% of the variance in anxiety scores.

3.3 Model 2 – does physical activity explain anxiety symptoms beyond screen time?

m2 <- lm(anxiety_score ~ screen_time + physical_activity, data = d)
summary(m2)
## 
## Call:
## lm(formula = anxiety_score ~ screen_time + physical_activity, 
##     data = d)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -31.81  -8.64   0.46   9.14  26.85 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         30.758      4.382    7.02  9.4e-11 ***
## screen_time          2.063      0.700    2.95  0.00377 ** 
## physical_activity   -1.582      0.441   -3.58  0.00047 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.7 on 137 degrees of freedom
## Multiple R-squared:  0.17,   Adjusted R-squared:  0.158 
## F-statistic:   14 on 2 and 137 DF,  p-value: 2.93e-06
confint(m2)
##                     2.5 %  97.5 %
## (Intercept)       22.0922 39.4228
## screen_time        0.6791  3.4473
## physical_activity -2.4552 -0.7095
anova(m1, m2)
## Analysis of Variance Table
## 
## Model 1: anxiety_score ~ screen_time
## Model 2: anxiety_score ~ screen_time + physical_activity
##   Res.Df   RSS Df Sum of Sq    F  Pr(>F)    
## 1    138 24219                              
## 2    137 22142  1      2077 12.8 0.00047 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Physical activity significantly improves the model (\(\Delta R^2\) = 0.078). Its coefficient is negative: each additional hour of weekly moderate-to-vigorous activity is associated with a 1.58-point decrease in predicted anxiety score, holding screen time constant. This is consistent with physical activity being protective against anxiety symptoms, independent of how much time an adolescent spends on screens.

3.4 Model 3 – is parental monitoring associated with anxiety, controlling for screen time and physical activity?

“Low” monitoring is coded as the reference category (0), “High” monitoring as 1. This choice does not change any conclusion about statistical significance or model fit; it only changes the sign and the wording of the monitoring coefficient.

m3 <- lm(anxiety_score ~ screen_time + physical_activity + monitoring, data = d)
summary(m3)
## 
## Call:
## lm(formula = anxiety_score ~ screen_time + physical_activity + 
##     monitoring, data = d)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -22.529  -6.484   0.084   5.562  20.342 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         36.596      3.073   11.91  < 2e-16 ***
## screen_time          2.674      0.487    5.48  2.0e-07 ***
## physical_activity   -1.599      0.306   -5.23  6.3e-07 ***
## monitoringHigh     -18.351      1.501  -12.23  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.81 on 136 degrees of freedom
## Multiple R-squared:  0.604,  Adjusted R-squared:  0.596 
## F-statistic: 69.3 on 3 and 136 DF,  p-value: <2e-16
confint(m3)
##                     2.5 %   97.5 %
## (Intercept)        30.519  42.6733
## screen_time         1.710   3.6376
## physical_activity  -2.203  -0.9938
## monitoringHigh    -21.319 -15.3823
anova(m2, m3)
## Analysis of Variance Table
## 
## Model 1: anxiety_score ~ screen_time + physical_activity
## Model 2: anxiety_score ~ screen_time + physical_activity + monitoring
##   Res.Df   RSS Df Sum of Sq   F Pr(>F)    
## 1    137 22142                            
## 2    136 10549  1     11593 149 <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Adolescents under high parental monitoring scored, on average, 18.35 points lower on the anxiety measure than those under low monitoring, holding screen time and physical activity constant. This is a large association, and adding monitoring more than tripled the variance explained by the model.

lm.beta(m3)
## 
## Call:
## lm(formula = anxiety_score ~ screen_time + physical_activity + 
##     monitoring, data = d)
## 
## Standardized Coefficients::
##       (Intercept)       screen_time physical_activity    monitoringHigh 
##                NA            0.3058           -0.2899           -0.6631

The two continuous predictors have fairly close standardized associations with anxiety score. lm.beta() also returns a number for monitoring, but, for the same reason discussed in the lab practical, it is not interpreted the same way: a dummy variable’s “standardized coefficient” reflects the arbitrary group split as much as the effect itself, so the unstandardized group difference above remains the right way to report the monitoring effect.

3.5 Checking the LINE assumptions

par(mfrow = c(2, 2))
plot(m3)

par(mfrow = c(1, 1))
durbinWatsonTest(m3)
##  lag Autocorrelation D-W Statistic p-value
##    1        -0.05477         2.085   0.638
##  Alternative hypothesis: rho != 0
shapiro.test(resid(m3))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(m3)
## W = 0.99, p-value = 0.9
vif(m3)
##       screen_time physical_activity        monitoring 
##             1.069             1.058             1.011

This is a useful point of contrast with the lab practical, where normality showed a mild, explainable departure: check whether all four LINE assumptions hold comfortably here, or whether a similar issue turns up. It is just as important to correctly conclude “no violation” as it is to correctly detect one.

d$resid_m3 <- resid(m3)
d %>% mutate(abs_resid = abs(resid_m3)) %>%
  arrange(desc(abs_resid)) %>%
  select(adolescent_id, screen_time, physical_activity, monitoring, anxiety_score, resid_m3) %>%
  head(1) %>%
  kable()
adolescent_id screen_time physical_activity monitoring anxiety_score resid_m3
92 5.4 4.1 High 3.6 -22.53 
cooks <- cooks.distance(m3)
sort(cooks, decreasing = TRUE)[1:5]
##     133      15      43      78      90 
## 0.04678 0.04535 0.04113 0.04077 0.03049

3.6 Model 4 – does the association between screen time and anxiety differ by monitoring level?

m4 <- lm(anxiety_score ~ screen_time * monitoring + physical_activity, data = d)
summary(m4)
## 
## Call:
## lm(formula = anxiety_score ~ screen_time * monitoring + physical_activity, 
##     data = d)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -21.88  -5.86   1.07   5.81  20.68 
## 
## Coefficients:
##                            Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                  33.207      3.650    9.10  1.1e-15 ***
## screen_time                   3.398      0.646    5.26  5.5e-07 ***
## monitoringHigh              -10.997      4.591   -2.40    0.018 *  
## physical_activity            -1.553      0.305   -5.10  1.2e-06 ***
## screen_time:monitoringHigh   -1.613      0.953   -1.69    0.093 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.75 on 135 degrees of freedom
## Multiple R-squared:  0.613,  Adjusted R-squared:  0.601 
## F-statistic: 53.4 on 4 and 135 DF,  p-value: <2e-16
confint(m4)
##                              2.5 %  97.5 %
## (Intercept)                 25.988 40.4250
## screen_time                  2.120  4.6752
## monitoringHigh             -20.077 -1.9176
## physical_activity           -2.156 -0.9505
## screen_time:monitoringHigh  -3.498  0.2708
anova(m3, m4)
## Analysis of Variance Table
## 
## Model 1: anxiety_score ~ screen_time + physical_activity + monitoring
## Model 2: anxiety_score ~ screen_time * monitoring + physical_activity
##   Res.Df   RSS Df Sum of Sq    F Pr(>F)  
## 1    136 10549                           
## 2    135 10329  1       219 2.87  0.093 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
b <- coef(m4)
slope_low  <- b[["screen_time"]]
slope_high <- b[["screen_time"]] + b[["screen_time:monitoringHigh"]]
c(Low = slope_low, High = slope_high)
##   Low  High 
## 3.398 1.784

Look carefully at the \(p\)-value for the interaction term before interpreting the two simple slopes above. If it is not below the conventional .05 threshold, the correct conclusion is that this sample does not provide statistically significant evidence that the association between screen time and anxiety differs by monitoring level, even if the two simple-slope point estimates look different from each other. The interaction test is exactly the test of whether that apparent difference is larger than what sampling variability alone would produce. In that case, the additive model (Model 3) remains the best-supported description of the data, and it is Model 3’s coefficients that should be the headline results; the simple slopes may be noted as a directional, non-significant observation for future research, but not reported as an established finding.

pa_mean <- mean(d$physical_activity)
newdat <- expand.grid(
  screen_time = seq(min(d$screen_time), max(d$screen_time), length.out = 100),
  monitoring = levels(d$monitoring),
  physical_activity = pa_mean
)
newdat$predicted <- predict(m4, newdata = newdat)

ggplot(newdat, aes(screen_time, predicted, color = monitoring)) +
  geom_line(linewidth = 1.2) +
  scale_color_manual(values = c("Low" = "grey40", "High" = ntnublue)) +
  labs(x = "Daily screen time (hours)", y = "Predicted anxiety score",
       title = "Predicted anxiety score by screen time and monitoring level",
       subtitle = "Physical activity held at its sample mean", color = "Monitoring")

3.7 Reporting the results

Daily screen time significantly predicted adolescent anxiety symptoms (Model 1). Physical activity added a significant further contribution, with more activity associated with lower anxiety (Model 2). Parental monitoring level was strongly associated with anxiety after accounting for both, with high-monitoring adolescents scoring lower on average (Model 3). A screen time by monitoring interaction was in the expected direction (a smaller screen-time slope under high monitoring) but should be reported as significant or non-significant strictly according to its own test (Model 4), not according to how different the two simple slopes look. Because screen time, physical activity, and monitoring level were all measured, not manipulated, these results describe associations only; they do not establish that screen time causes anxiety symptoms, or that increasing parental monitoring would reduce them, since unmeasured factors (for example, an adolescent’s baseline anxiety influencing both their screen use and how closely a parent chooses to monitor it) could account for part or all of the observed associations.

4 Solutions

Expected answers to the Questions above, reusing the models fitted in the Guided Walkthrough section.

  1. \(N\) = 140 (65 High monitoring, 75 Low monitoring). See the descriptives chunk above for means and SDs by group. \(r\)(screen_time, physical_activity) = -0.234: small enough that entering both predictors together should not create serious collinearity (confirmed by the VIF check below), but not negligible, so each coefficient should be read as controlling for the other.

  2. \(\widehat{anxiety\_score}\) = 20.26 + 2.65 \(\times\) screen_time. See summary(m1) above for \(SE\), \(t\), \(df\), and \(p\). \(R^2\) = 0.092. Each additional hour of daily screen time is associated with a 2.65-point increase in predicted anxiety score, and the effect is significant.

  3. \(R^2\) = 0.17, \(\Delta R^2\) = 0.078. See anova(m1, m2) above for the full test (residual sums of squares, \(df\), \(F\), \(p\)). The physical-activity coefficient is negative (-1.58), consistent with physical activity being protective against anxiety, independent of screen time.

  4. “Low” monitoring is the reference category (0); this choice changes only which coefficients are printed and their sign, not the model’s fit or its substantive conclusions. Monitoring coefficient = -18.35 (High vs. Low). See summary(m3) and anova(m2, m3) above for the test statistics and \(\Delta R^2\) (= 0.435).

  5. See the lm.beta(m3) output above. The two continuous predictors have fairly close standardized associations; the number lm.beta() returns for the monitoring dummy is not interpreted the same way, for the reason given in the Guided Walkthrough (it reflects the group split, not a meaningful “amount” of the categorical predictor).

  6. See the diagnostic plots and test statistics above. Compare each printed test’s \(p\)-value against .05 and each plot against the pattern expected under no violation; contrast explicitly with the lab practical’s mild normality departure.

  7. See the largest residual chunk above: report the identified adolescent_id and compare its Cook’s distance against the conventional concern threshold of 1.0 (and against the other largest values printed) to judge influence, not the residual alone.

  8. See summary(m4) and anova(m3, m4) above for the interaction coefficient and its test.

  9. If the interaction \(p\)-value from Question 8 is not below .05 (as is the case for this dataset), the correct conclusion is that the data do not provide significant evidence that monitoring moderates the screen-time effect. It is not appropriate to present the two simple slopes as a confirmed pattern; the additive model (Model 3) remains the best-supported description, and any mention of the directionally different slopes must be clearly flagged as non-significant and exploratory, not reported as a finding.

  10. See the Reporting the results paragraph above for a model answer, including the explicit non-causal caveat given that screen time, physical activity, and monitoring were all measured rather than manipulated.

Common errors and points to watch for

sessionInfo()
## R version 4.4.2 (2024-10-31)
## Platform: aarch64-apple-darwin20
## Running under: macOS 26.6.2
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib 
## LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## time zone: Europe/Oslo
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] knitr_1.51      car_3.1-5       carData_3.0-6   lm.beta_1.7-3   lubridate_1.9.5 forcats_1.0.1   stringr_1.6.0  
##  [8] dplyr_1.2.0     purrr_1.2.1     readr_2.2.0     tidyr_1.3.2     tibble_3.3.1    ggplot2_4.0.2   tidyverse_2.0.0
## [15] pacman_0.5.1   
## 
## loaded via a namespace (and not attached):
##  [1] gtable_0.3.6       jsonlite_2.0.0     compiler_4.4.2     tidyselect_1.2.1   jquerylib_0.1.4    scales_1.4.0      
##  [7] yaml_2.3.12        fastmap_1.2.0      R6_2.6.1           labeling_0.4.3     generics_0.1.4     Formula_1.2-5     
## [13] tzdb_0.5.0         bslib_0.10.0       pillar_1.11.1      RColorBrewer_1.1-3 rlang_1.1.7        stringi_1.8.7     
## [19] cachem_1.1.0       xfun_0.57          sass_0.4.10        S7_0.2.1           otel_0.2.0         timechange_0.4.0  
## [25] cli_3.6.5          withr_3.0.2        magrittr_2.0.4     digest_0.6.39      grid_4.4.2         xtable_1.8-8      
## [31] rstudioapi_0.18.0  hms_1.1.4          lifecycle_1.0.5    vctrs_0.7.2        evaluate_1.0.5     glue_1.8.0        
## [37] farver_2.1.2       abind_1.4-8        rmarkdown_2.30     tools_4.4.2        pkgconfig_2.0.3    htmltools_0.5.9