Regression analysis in psychological research

Regression vs. correlation?

Correlation is used to test whether two variables covary, the strength of the relationship, and the direction of the association. A correlation calculation will generate a P-value and a correlation coefficient (r). By comparison, regression will generate the slope and intercept for a best-fit line that can be used to predict unknown values for the dependent variable.

What percentage of depression is not associated with Facebook usage?

The coefficient of determination (r2) is 0.661, which means that 66.1% of the variance in depression is due to the amount of time spent on Facebook; therefore, 33.9% of the variation in depression cannot be explained by time spent on Facebook.

Q3: Variables that could be contributing to the variance not explained by time spent on Facebook?

The unexplained variance in depression scores is the amount of error between measured levels of depression for a study subject and what was predicted by the regression line. This error is due to other variables, possibly naturally-occurring variation. Natural variation in depression could be explained in part by genetic differences or environmental factors like early life experiences (Klengel & Binder, 2013). The amount of variation explained by genetic factors is represented by the standard error of estimate, but the magnitude of the contribution, if it exists, is unknown.

Q4: Confident in the predictive power of the regression equation?

Yes, the significance of the correlation suggests that the chances of making a prediction error are less than 1 in 1000 (p < 0.000). Stated another way, the amount of time spent on Facebook would correctly predict depression scores more than 999 times out of a 1,000.

Q5: Results interpretation

Despite the relatively small sample size (N = 50) the strength of the relationship between depression scores and time spent on Facebook was fairly strong (r = 0.813, p < 0.000). This relationship is positive, which suggests that more time on Facebook is associated with higher levels of depression. What cannot be determined is whether there is a causal relationship, since time on Facebook could be contributing to depression or vice versa, or the relationship could be indirect. The only conclusion that can be drawn is that time on Facebook is a good predictor of depression.

Q6: Calculate depression score based on 120 minutes per week Facebook time.

From the table of coefficients the slope of the regression line is 0.135 and the intercept is 3.061. The depression score predicted by 120 minutes of Facebook time is therefore 3.061 + 0.135*120 = 19.261. A quick comparison with the group means reveals this answer is probably correct.

Assignment 2.2: One-Way ANOVA

Q1: Complete the table:

Groups: Drug administration daily, weekly, or null

12 rats per group

Outcome measure is the number of food pellets consumed per month

Source

SS

df

MS

F

Between Groups (Treatment)

24

2

12

6

Within Groups (Error)

66

33

2

Total

78

35

Q2: Find the critical F. value for an alpha of 0.05.

Fcrit (2, 33) = 3.285 for the above data.

Q3: Evaluate the above findings.

The Fcrit (2, 33) value of 3.285 is significantly below the F. value of 6, therefore, the results are significant using an alpha of 0.05. This result forces us to reject the null hypothesis for this experiment, which would be that the drug has no effect on the number of pellets consumed during the month.

Q4: How are the t-test and One-way ANOVA similar? Given an example where both would work.

Mathematically, there is no difference between a t-test and a One-way ANOVA if only two groups are compared. An example of a comparison where both tests could be used would be the number of hours of sleep students obtain the night before an exam and the resulting exam scores.

Q5: When would a One-way ANOVA be more appropriate than a t-test?

The t-test should not be used to test for a significant association between more than two groups. The above example has three groups, so the appropriate test is ANOVA if assumptions of normality and homoscedasticity are met.

Assignment 2.3: ANOVA and Post-hoc Tests

Q1: When is it appropriate to conduct post-hoc analysis after an ANOVA has been completed?

When the F. statistic exceeds the Fcrit value, the ANOVA test has found a significant difference in the means for at least two of the groups being compared. What the ANOVA does not do is identify which between group comparisons were significant. For this reason a post-hoc analysis must be performed to identify which group means contributed to a significant ANOVA result.

Q2: What is the Family-wise error rate and how does post-hoc analysis affect it?

When two groups are compared using the t-test the chances of getting a type I error is usually set to an alpha of 0.05 or 0.01. If three groups were compared, the chances of getting a type I error is almost doubled. As the number of groups being compared is increased, so is the chance of getting a type I error. This is called the family-wise error rate. Post-hoc tests attempt to control for family-wise error, but to varying degrees. For example, the Fisher's least significant difference is appropriate for comparing three means, but including more groups will inflate the family-wise error beyond the chosen alpha level. By comparison, the Tukey's test is designed to keep alpha at the desired level regardless of the number of groups compared.