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CASE STUDY: CORRELATION AND BIVARIATE REGRESSION TEMPLATE
I. Correlation Research question: “ Is mean heart rate during exercise correlated to body weight? ” Assumptions Testing
- Data level of measurement – what was the level of measurement for the data used in this case study (ratio, interval, etc.)? ______Ratio________________________________________________________
- Would the amount of skewness and kurtosis in these variables affect the analysis? How do you know? Give numerical values to support your conclusion. For mean heart rate, the skewness value is 0.366. This value is close to zero, which means the data is fairly symmetrical. The kurtosis value is -0.519, suggesting that the data distribution is not too different from a normal distribution. These values indicate that the data for mean heart rate is approximately normal, and this should not affect our analysis much. For body weight, the skewness value is 1.344. This higher value means the data is skewed to the right, with a longer tail on the right side. The kurtosis value is 1.620, indicating that the data has a sharper peak and heavier tails compared to a normal distribution. These values suggest that the body weight data is not normally distributed, which might affect analyses that assume normality, like correlation and regression (Table 1).
**Table 1. Table with skewness and kurtosis** Paste table below:
Statistics
Body_weight_ Kg Mean_heart_r ate_BPM N Valid 20 20 Missing 0 0 Skewness 1.344. Std. Error of Skewness
Kurtosis 1.620 -. Std. Error of Kurtosis .992.
- Was the assumption of normality met and how do you know? For Mean heart rate the Kolmogorov-Smirnov test gave a statistic of 0.091 and a p-value of 0.200. The Shapiro-Wilk test gave a statistic of 0.964 and a p-value of 0.621. Both p-values are higher than 0.05. This means we do not have enough evidence to say the data is different from a normal distribution. For body weight , the Kolmogorov-Smirnov test gave a statistic of 0.208 and a p-value of 0.024. The Shapiro-Wilk test gave a statistic of 0.861 and a p-value of 0.008. Both p-values are lower than 0.05 (Table 2). This means we have enough evidence to say the data is different from a normal distribution. So, we can say that the body weight data is not normally distributed.
**Table 2. Tests of normality** Paste table below:
Tests of Normality
Kolmogorov-Smirnova^ Shapiro-Wilk Statistic df Sig. Statistic df Sig. Mean_heart_rate_B PM
.091 20 .200*^ .964 20.
Body_weight_Kg .208 20 .024 .861 20. *. This is a lower bound of the true significance. a. Lilliefors Significance Correction
Figure 2. Normal Q-Q plot for each variable Paste figure below:
Statistical Analysis Table 4. Descriptives statistics table Paste table below:
Descriptives
Statistic Std. Error Mean_heart_rate_BP M Mean 132.65 3. 95% Confidence Interval for Mean Lower Bound
Upper Bound
5% Trimmed Mean 132. Median 133. Variance 250. Std. Deviation 15. Minimum 108 Maximum 165 Range 57 Interquartile Range 25 Skewness .366. Kurtosis -.519. Body_weight_Kg Mean 85.845 2. 95% Confidence Interval for Mean Lower Bound
Upper Bound
5% Trimmed Mean 84. Median 81. Variance 155. Std. Deviation 12. Minimum 71. Maximum 120. Range 49. Interquartile Range 19. Skewness 1.344. Kurtosis 1.620.
and relate the findings to exercise physiology knowledge. Use references if needed. (Hint- does this study “make sense?” What could influence results?) The research question was: “Is mean heart rate during exercise correlated to body weight?” The Pearson correlation value (r) was -0.411, and the p-value was 0.072. This p-value is greater than the standard significance level of 0.05, indicating that the correlation was not statistically significant. The relationship between the two variables was negative. This means that as body weight increased, mean heart rate tended to decrease slightly. However, the correlation was weak, as the value of -0.411 is relatively low and close to zero. The findings of this analysis suggest that there is no significant relationship between body weight and mean heart rate during exercise in this sample. This weak and non- significant correlation indicates that other factors may play a more crucial role in determining heart rate during exercise, such as fitness level, type of exercise, and individual physiological differences. II. Bivariate Regression Research question: “ Can heart rate during exercise be predicted by using body weight as a predicting variable and creating a linear regression equation?” (THIS SECTION USES the SAME DATA and will replicate results from above, other than the REGRESSION Section) Assumptions Testing
- Data level of measurement- what was the level of measurement for the data used in this case study (ratio, scale etc.)? _____ Ratio _________________________________________________________
- Would the amount of skewness and kurtosis in these variables affect the analysis? How do you know? Give numerical values to support your conclusion. Skewness and kurtosis values help determine if the data is normally distributed. For Mean heart rate, the skewness is 0.366 and the kurtosis is -0.519. These values are close to zero, indicating that the distribution is approximately normal, which is good for regression analysis. For Body weight, the skewness is 1.344 and the kurtosis is 1.620. These higher values indicate a skewed and peaked distribution, which suggests that the data is not
normally distributed. Non-normality in the predictor variable (body weight) can affect the results of the regression analysis, potentially making it less reliable.
Table 1. Table with skewness and kurtosis Paste table below:
Statistics
Body_weight_ Kg Mean_heart_r ate_BPM N Valid 20 20 Missing 0 0 Skewness 1.344. Std. Error of Skewness
Kurtosis 1.620 -. Std. Error of Kurtosis .992.
- Was the assumption of normality met and how do you know?
**Table 2. Tests of normality** Paste table below:
Tests of Normality
Kolmogorov-Smirnova^ Shapiro-Wilk Statistic df Sig. Statistic df Sig. Mean_heart_rate_B PM
.091 20 .200*^ .964 20.
Figure 2. Normal Q-Q plot for each variable Paste figure below:
Statistical Analysis Table 4. Descriptives statistics table Paste table below:
Descriptives
Statistic Std. Error Mean_heart_rate_BP M Mean 132.65 3. 95% Confidence Interval for Mean Lower Bound
Upper Bound
5% Trimmed Mean 132. Median 133.
Variance 250. Std. Deviation 15. Minimum 108 Maximum 165 Range 57 Interquartile Range 25 Skewness .366. Kurtosis -.519. Body_weight_Kg Mean 85.845 2. 95% Confidence Interval for Mean Lower Bound
Upper Bound
5% Trimmed Mean 84. Median 81. Variance 155. Std. Deviation 12. Minimum 71. Maximum 120. Range 49. Interquartile Range 19. Skewness 1.344. Kurtosis 1.620. Table 5. Correlations table Paste table below:
Correlations
ANOVA
a Model Sum of Squares df Mean Square F Sig. 1 Regression 803.927 1 803.927 3.654 .072b Residual 3960.623 18 220. Total 4764.550 19 a. Dependent Variable: Mean_heart_rate_BPM b. Predictors: (Constant), Body_weight_Kg Table 8. The coefficients table Use this table to type out the regression equation for this data. Paste table below:
Coefficients
a Model Unstandardized Coefficients Standardized Coefficients t Sig. Collinearity Statistics B Std. Error Beta Toleranc e VIF 1 (Constant) 177.374 23.632 7.506 <. Body_weight_ Kg
a. Dependent Variable: Mean_heart_rate_BPM
Heart Rate (BPM)=177.374−0.521×Body Weight (Kg) Write-up- Use these questions as a guide to write an abstract (paragraph). Use past tense. Use the following questions as a guide:
- What was the research question?
- What was the Pearson correlation value (r), and was it statistically significant? (Use “sig” reported in SPSS. That is the p -value, and compare to alpha to determine significance)
- Was the relationship positive or negative and what was the (magnitude) strength of the relationship (use r/Pearson)?
- What was the regression equation for these data? Consider y=mx + b where y = HR.
- What are your thoughts on the finding of this analysis, and what are the practical application(s) of this finding? Elaborate on what the study means, why, how you know, and relate the findings to exercise physiology knowledge. Use references if needed. (Hint- does this study “make sense?” What could influence results?)
