5.5.2. Perceived Intensity as Dependent Variable
In the second part of the DiD regression analysis we investigate the impact of the intervention on the perceived training intensity. The training intensity represents a scale between 0 and 10 with higher values standing for a more intense training. To analyze the variable intensity, we conduct the same steps as in the previous section. First, the analysis focuses on the impact of the intervention in the individual time periods: on the one hand, from the baseline test to the first post-test, and on the other hand, from the first post-test to the second post-test. In the next step, we run a DiD regression on the entire trial. Finally, we test a dynamic DiD regression model that accounts for both time periods in one model and individual treatment effects for each time period. Similar to the analysis of the injury score, we show two models for each step, of which the first one will be a simple regression model with the time variable, the group variable and the interaction term. The second model will include additional variables as covariates.
As the results in regression model 8 shows, the model fits the data quite well as the adjusted R-squared value of 0.710 suggests. First, the intercept is at 3.964, which in a DiD regression model represents the perceived training intensity at the baseline test when all the independent variables are at their baseline level. The time variable shows that the intensity in both groups increased over time, with a coefficient of 2.145 and high statistical significance (p < 0.01), which means that the dependent variable training intensity is on average 2.15 units higher compared to the baseline test. Most importantly, the interaction term shows a coefficient of 0.802 and a high statistical significance (p < 0.05), which means the changes from the baseline test to the first post-test were 0.8 greater in the control group than in the intervention group. In other words, intensity increased in both groups over time, but the change in intensity of the intervention group was lower than in the control group. The control group tends to have a lower intensity, but this gap widens at the first post-test, as shown by the interaction term.
In addition to the simple DiD regression model, we now conduct a multivariate DiD regression model, which includes additional covariates. We decided to prioritize the variable weight as covariate because the variable had the highest correlation with the intensity, as in section 3.4 was shown. Furthermore, the variables age and bicycle type mountain bike were excluded from the model because of a variety of reasons. First, none of those two variables showed any sign of association with the perceived training intensity in the correlation analysis, even though we expected a correlation, at least between age and training intensity. Second, the inclusion of the two variables significantly reduces the accuracy and fitness of the regression model. Other variables, which we excluded from the model, were the variables BMI and height, even though we found a moderate positive statistically significant correlation between these two variables and the perceived training intensity. The reason for this is that the independent variables in a regression model should not correlate with each other due to the problem of multicollinearity.
As can be derived from the results in
Table 14, the multivariate DiD model has an adjusted R-squared of 0.734, which means that 73.4 % of the variability in the dependent variable can be explained by the model. Compared to the simple regression in our first model, R-squared improved, but only slighty by one percentage point.
The multivariate model reveals results for the coefficient and p-value of the time variable (2.145, p=0.000), which are like the previous model. Both groups experienced an increase in intensity over time. Looking at the treatment effect variable allows us to conclude that the impact of the intervention on the perceived training intensity remains strong and statistically significant (0.80, p = 0.039), even after accounting for weight. This term tells us that the change in intensity from the baseline test to the first post-test was 0.80 units higher in the control group than the change observed in the intervention group. In other words, while both groups experienced an increase in intensity over time, the increase was significantly greater in the control group. The rejection of the null hypothesis for the interaction term implies a statistically meaningful difference in how intensity changed over time between the two groups.
The coefficient for body weight is positive and statistically significant (0.0326, p = 0.011), indicating that for each additional kilogram of body weight, the intensity increases by about 0.033 units, while the variable remains constant. This suggests that heavier individuals tend to report a slightly higher training intensity. This effect is relatively small in absolute terms, but it is statistically significant (p = 0.011), which means we can reject the null hypothesis that weight has no impact on intensity. Individuals with higher body weight tend to perceive their training as more intense, which could reflect differences in physical exertion or perceived effort due to body composition. This relationship holds regardless of time point or group, since the model assumes the effect of weight is constant across both groups and time.
In regression model 10 in
Table 15 the goal was to estimate the effect of the predictors time, group and the interaction term on the perceived training intensity for the second time-period, which started with the first post-test and ended with the second post-test. As can be seen in the R-squared value, the model explains about 38% of the variation in the perceived training intensity, which means that the model has a moderate explanatory power. For the time variable the model estimated a coefficient of 0.81, which indicates that the intensity increased in both groups – a result that is highly significant with p = 0.002. Regarding the group variable, we did not find a statistically significant difference between the groups at the first post-test (-0.089, p = 0.726). The interaction effect indicates that the change in the control group is slightly greater than in the intervention group. However, the p-value for the interaction term is 0.130, which means that it is not statistically significant at conventional threshold. In other words, there is no strong statistical evidence suggesting that the intervention group changed differently over time compared to the control group, nor that the two groups differed significantly at any specific time point.
In
Table 16 we updated the model by including the variable weight as covariate in addition to the time variable, group variable and the interaction term. Compared to the previous model without the covariate, the R-squared has increased slightly from 0.376 to 0.393, indicating a marginal improvement in the model’s explanatory power. However, this increase is modest and given that body weight is not a statistically significant predictor, the added complexity does not lead a substantial gain in the accuracy of the model. The interaction term, which is our most crucial variable, has a positive coefficient of 0.548, which suggests that the control group may have experience a slightly greater increase in intensity over time compared to the intervention group, but this effect is not statistically significant at the p < 0.05 or p < 0.01 level (p-value = 0.127). The new covariate weight has a coefficient of 0.015, suggesting that for each additional kilogram of body weight, training intensity increases by approximately 0.015 units.
Table 13.
Regression model 9 estimates the impact of the intervention from the baseline test to the first post-test on the perceived training intensity (Source: own illustration/Python).
Table 13.
Regression model 9 estimates the impact of the intervention from the baseline test to the first post-test on the perceived training intensity (Source: own illustration/Python).
| Variable |
Coefficient |
Std. Error |
t-value |
p-value |
95% confidence interval |
| Intercept |
3.964 |
0.161 |
25.571 |
0.000 |
[3.642, 4.287] |
| Time |
2.954 |
0.228 |
12.948 |
0.000 |
[2.499, 3.410] |
| Group (control group) |
-0.892 |
0.232 |
-3.851 |
0.000 |
[-1.354, -0.429] |
| Interaction term |
1.350 |
0.327 |
4.124 |
0.000 |
[0.697, 2.004] |
Table 14.
Regression model 10 estimates the impact of the intervention from the baseline test to the first post-test on the perceived training intensity with additional covariates (Source: own illustration/Python).
Table 14.
Regression model 10 estimates the impact of the intervention from the baseline test to the first post-test on the perceived training intensity with additional covariates (Source: own illustration/Python).
| Variable |
Coefficient |
Std. Error |
t-value |
p-value |
95% confidence interval |
| Intercept |
1.378 |
1.008 |
1.367 |
0.176 |
[-0.636, 3.392] |
| Time |
2.145 |
0.266 |
8.059 |
0.000 |
[1.613, 2.677] |
| Group (control group) |
-0.708 |
0.279 |
-2.537 |
0.014 |
[-1.265, -0.151] |
| Interaction term |
0.803 |
0.382 |
2.015 |
0.039 |
[0.040, 1.565] |
| Weight |
0.033 |
0.012 |
2.611 |
0.011 |
[0.008, 0.057] |
Table 15.
Regression model 11 estimates the impact of the intervention from the first post-test to the second post-test on the perceived training intensity (Source: own illustration/Python).
Table 15.
Regression model 11 estimates the impact of the intervention from the first post-test to the second post-test on the perceived training intensity (Source: own illustration/Python).
| Variable |
Coefficient |
Std. Error |
t-value |
p-value |
95% confidence interval |
| Intercept |
6.109 |
0.176 |
34.76 |
0.000 |
[5.758, 6.460] |
| Time |
0.809 |
0.249 |
3.256 |
0.002 |
[0.313, 1.306] |
| Group (control group) |
-0.089 |
0.252 |
-0.352 |
0.726 |
[-0.592, -0.415] |
| Interaction term |
0.548 |
0.357 |
1.535 |
0.130 |
[-0.165, 1.260] |
Table 16.
Regression model 12 estimates the impact of the intervention from the first post-test to the second post-test on the perceived training intensity with additional covariates (Source: own illustration/Python).
Table 16.
Regression model 12 estimates the impact of the intervention from the first post-test to the second post-test on the perceived training intensity with additional covariates (Source: own illustration/Python).
| Variable |
Coefficient |
Std. Error |
t-value |
p-value |
95% confidence interval |
| Intercept |
4.891 |
0.936 |
5.223 |
0.000 |
[3.021, 6.761] |
| Time |
0.809 |
0.247 |
3.275 |
0.002 |
[0.316, 1.303] |
| Group (control group) |
-0.002 |
0.259 |
-0.009 |
0.993 |
[-0.520, 0.515] |
| Interaction term |
0.548 |
0.355 |
1.544 |
0.127 |
[-0.161, 1.256] |
| Weight |
0.015 |
0.012 |
1.324 |
0.190 |
[0.008, 0.038] |
The regression model 12 in
Table 17 provides an analysis of the relationship between the training intensity, on the one hand, and group membership and the interaction between time and group, on the other, starting with the baseline test and ending with the second post-test. In other words, this DiD regression model estimates the impact of the intervention over the entire trial. Overall, the model has a relatively high R-squared value of 0.885, which means that 88.5% of the variance in the training intensity can be explained by the independent variables included in the model. The adjusted R-squared value is 0.879, which adjusts for the number of predictors in the model and is still quite high, indicating a good fit.
Looking at the individual variables, we can conclude that the intensity of the training increased from the baseline test to the second post-test independent from group affiliation. With a coefficient of 2.954 and a p-value of 0.000 we can conclude with a relatively high certainty that on average the training intensity increased by 2.954 units from the baseline test to the second post-test. The interaction term between time and group is 1.3503, which means that the difference in training intensity between the intervention and control group is greater at the second post-test compared to the baseline test. To be more specific, the difference in training intensity between the intervention and control group at the second post-test is 1.35 units greater than at the baseline test. With a p-value of 0.000 this effect shows a high statistical significance. The interaction between time and group further strengthens the assumption, that the effect of time on training intensity is greater for the control group than for the intervention group.
In regression model 13 we extend the previous model with the covariate weight. As the summary statistics of the regression model shows, R-squared slightly increases with the additional covariate, but only by 0.7 percentage points. The model confirms most of the findings in the previous model, indicating a strong and statistically significant impact of the intervention on the perceived training intensity. All variables show a high level of statistical significance, the effect of the intervention expressed in the interaction term remains strong and statistically significant, even after controlling for weight.
Table 18.
Regression model 14 estimates the impact of the intervention from the baseline test to the second post-test on the perceived training intensity with additional covariates (Source: own illustration/Python).
Table 18.
Regression model 14 estimates the impact of the intervention from the baseline test to the second post-test on the perceived training intensity with additional covariates (Source: own illustration/Python).
| Variable |
Coefficient |
Std. Error |
t-value |
p-value |
95% confidence interval |
| Intercept |
2.158 |
0.841 |
2.567 |
0.013 |
[0.479, 3.836] |
| Time |
2.954 |
0.222 |
13.314 |
0.000 |
[2.511, 3.398] |
| Group (control group) |
-0.763 |
0.233 |
-3.281 |
0.002 |
[-1.228, -0.299] |
| Interaction term |
1.350 |
0.318 |
4.241 |
0.000 |
[0.714, 1.986] |
| Weight |
0.023 |
0.010 |
2.188 |
0.032 |
[0.002, 0.044] |
Whereas the previous models estimated the effects of the intervention and additional covariates on the intensity from one point in time to another point in time, the last two DiD regression models analyzed in the following paragraphs represent dynamic models that provide a more nuanced picture of how training intensity evolved over time by incorporating treatment effects for each time period.
The dynamic DiD regression model 14 explains a substantial proportion of the variance in intensity with an R-squared of 0.815 and an adjusted R-squared of 0.805, which indicates a strong fit of the model to the data points. The intercept, which represents the mean intensity in the control group at the baseline test, is estimated at 3.96. Compared to this baseline, the training intensity increases significantly over time. At the first post-test, intensity increases by about 2.145 units and at the second post-test, it increases even more by about 2.954 units – both effects are statistically significant with p-values < 0.001. This suggests that the control group experienced substantial increases in intensity over time. Most importantly, the interaction terms, which represent the additional change in the intervention group, are both aligned with our previous models. The coefficient for the first treatment effect is about 0.803 and significant at the 5% level (p = 0.029), while the coefficient for the second treatment effect is even stronger 1.350 and a p-value < 0.001 (p = 0.000). As the results indicate, both groups showed an increase in the perceived training intensity, however, the increases in the intervention group were consistently and significantly lower than the increases in intensity in the control group.
Finally, we added the weight as covariate to the model in addition to the time variable, group affiliation and treatment indicators. Like the previous model, the dynamic model shows a strong overall fit with an R-squared value of 0.826 and an adjusted R-squared value of 0.815, which means that about 81.5-82.6 % of the variance in the intensity can be explained by the model.
The coefficients for the time variables show that the perceived intensity increases significantly in both time periods. To be more specific, the intensity rises by around 2.15 units in the first period and by about 2.95 units at the second post-test. As for our most crucial independent variable, the interaction terms are both positive and statistically significant. This means that the control group experienced a significantly greater increase in intensity at both post-treatment time points compared to the intervention group. The treatment effect at the first post-test is estimated at about 0.80 units and increases further to 1.35 units at the second post-test. Finally, the body weight variable is positively and significantly associated with intensity. The coefficient of 0.024 suggests that for every additional kilogram of body weight, intensity increases by about 0.02 units, holding all other variables constant. This further supports the inclusion of weight as a meaningful covariate in the model. To summarize the model results, the results confirm the earlier findings that the implementation of the intervention resulted in lower increase in the intervention group compared to the control group – a trend that remains after controlling for body weight.
Table 19.
Regression model 15 shows a dynamic regression model and estimates the impact of the intervention on the intensity from the baseline test to the first post-test and from the first post-test to the second post-test (Source: own illustration/Python).
Table 19.
Regression model 15 shows a dynamic regression model and estimates the impact of the intervention on the intensity from the baseline test to the first post-test and from the first post-test to the second post-test (Source: own illustration/Python).
| Variable |
Coefficient |
Std. Error |
t-value |
p-value |
95% confidence interval |
| Intercept |
3.964 |
0.178 |
22.223 |
0.000 |
[3.610, 4.318] |
| Time (P1) |
2.145 |
0.252 |
8.502 |
0.000 |
[1.644, 2.646] |
| Time (P2) |
2.954 |
0.252 |
11.711 |
0.000 |
[2.454, 3.455] |
| Group (control group) |
-0.892 |
0.256 |
-3.483 |
0.001 |
[-1.399, -0.384] |
| Interaction term (P1) |
0.803 |
0.362 |
2.217 |
0.029 |
[0.084, 1.521] |
| Interaction term (P2) |
1.350 |
0.362 |
-3.730 |
0.000 |
[0.632, 2.069] |
Table 20.
Regression model 16 shows a dynamic regression model and estimates the impact of the intervention and additional covariates from the baseline test to the first post-test and from the first post-test to the second post-test on the perceived intensity (Source: own illustration/Python).
Table 20.
Regression model 16 shows a dynamic regression model and estimates the impact of the intervention and additional covariates from the baseline test to the first post-test and from the first post-test to the second post-test on the perceived intensity (Source: own illustration/Python).
| Variable |
Coefficient |
Std. Error |
t-value |
p-value |
95% confidence interval |
| Intercept |
2.094 |
0.767 |
2.730 |
0.008 |
[0.572, 3.616] |
| Time (P1) |
2.145 |
0.246 |
8.726 |
0.000 |
[1.657, 2.633] |
| Time (P2) |
2.954 |
0.246 |
12.018 |
0.000 |
[2.467, 3.442] |
| Group (control group) |
-0.759 |
0.255 |
-2.976 |
0.004 |
[-1.265, -0.253] |
| Interaction term (P1) |
0.803 |
0.353 |
2.276 |
0.025 |
[0.103, 1.503] |
| Interaction term (P2) |
1.350 |
0.353 |
-3.828 |
0.000 |
[0.650, 2.050] |
| Weight |
0.024 |
0.009 |
2.504 |
0.014 |
[0.005, 0.042] |