Running a Simple Regression
Nathan Clearwaters
2025-06-15
1 Loading Libraries
#install.packages("broom")
#install.packages("ggplot2")
library(psych) # for the describe() command
library(broom) # for the augment() command
library(ggplot2) # to visualize our results##
## Attaching package: 'ggplot2'## The following objects are masked from 'package:psych':
##
## %+%, alpha2 Importing Data
# For HW, import the dataset you cleaned previously, this will be the dataset you'll use throughout the rest of the semester
d <- read.csv(file="Data/projectdata.csv", header=T)3 State Your Hypothesis
We hypothesize that people’s reported level of neuroticism will significantly predict their level of intolerance of uncertainty, and that the relationship will be positive. This means that as people report higher levels of neuroticism, they will also report higher levels of intolerance of uncertainty.
My independent variable (the one doing the predicting) is: neuroticism My dependent variable (the one being predicted) is: intolerance of uncertainty
4 Check Your Variables
# you only need to check the variables you're using in the current analysis
# although you checked them previously, it's always a good idea to look them over again to be sure that everything is correct
str(d)## 'data.frame': 1326 obs. of 7 variables:
## $ X : int 1 321 401 469 1390 2183 2247 2482 2526 2609 ...
## $ gender : chr "female" "male" "female" "female" ...
## $ treatment: chr "no psychological disorders" "not in treatment" "not in treatment" "in treatment" ...
## $ big5_open: num 5.33 4 6 5 3 ...
## $ big5_con : num 6 4.33 5 5.67 3.33 ...
## $ big5_neu : num 6 3.67 3 4.33 3.33 ...
## $ iou : num 3.19 2.48 2.81 2.59 1.48 ...# you can use the describe() command on an entire dataframe (d) or just on a single variable
describe(d)## vars n mean sd median trimmed mad min max range
## X 1 1326 4825.98 2595.50 5092.50 4911.18 3286.18 1 8858 8857
## gender* 2 1326 1.41 0.82 1.00 1.24 0.00 1 4 3
## treatment* 3 1326 2.72 1.09 3.00 2.58 1.48 1 6 5
## big5_open 4 1326 5.21 1.15 5.33 5.30 0.99 1 7 6
## big5_con 5 1326 4.80 1.19 4.67 4.84 1.48 1 7 6
## big5_neu 6 1326 4.44 1.51 4.67 4.50 1.48 1 7 6
## iou 7 1326 2.61 0.92 2.48 2.56 1.04 1 5 4
## skew kurtosis se
## X -0.22 -1.22 71.28
## gender* 1.66 1.12 0.02
## treatment* 1.31 2.29 0.03
## big5_open -0.74 0.42 0.03
## big5_con -0.26 -0.30 0.03
## big5_neu -0.33 -0.74 0.04
## iou 0.46 -0.61 0.03# next, use histograms to examine your continuous variables
hist(d$big5_neu)
hist(d$iou)
# last, use scatterplots to examine your continuous variables together
# Remember to put INDEPENDENT VARIABLE FIRST, so that it goes on the x-axis
plot(d$big5_neu, d$iou)
5 Run a Simple Regression
# to calculate standardized coefficients for the regression, we have to standardize our IV
d$big5_neu_std <- scale(d$big5_neu, center=T, scale=T)
# use the lm() command to run the regression
# dependent/outcome variable on the left of the ~, standardized independent/predictor variable on the right.
reg_model <- lm(iou ~ big5_neu_std , data = d)
# NO PEEKING AT YOUR MODEL RESULTS YET!6 Check Your Assumptions
6.1 Simple Regression Assumptions
- Should have two measurements for each participant
- Variables should be continuous and normally distributed
- Relationship between the variables should be linear
- Outliers should be identified and removed
- Residuals should be approx. normal and have constant variance NOTE: We will NOT be evaluating whether our data meets this last assumption in this lab/homework.
6.2 Create plots and view residuals
# Create Plots
model.diag.metrics <- augment(reg_model)
# View Raw Residuals Plot
# NOTE: only replace the variables in 3 places in this line of code
ggplot(model.diag.metrics, aes(x = big5_neu_std, y = iou)) +
geom_point() +
stat_smooth(method = lm, se = FALSE) +
geom_segment(aes(xend = big5_neu_std, yend = .fitted), color = "red", size = 0.3)## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.## `geom_smooth()` using formula = 'y ~ x'
6.3 Check linearity with Residuals vs Fitted plot
The plot below shows the residuals for each case and the fitted line. The red line is the average residual for the specified point of the dependent variable. If the assumption of linearity is met, the red line should be horizontal. This indicates that the residuals average to around zero. You can see that for this lab, the plot shows some non-linearity because there are more data points below the regression line than there are above it. Thus, there are some negative residuals that don’t have positive residuals to cancel them out. However, a bit of deviation is okay – just like with skewness and kurtosis with non-normality – there is a range of acceptability that we can work in before non-linearity becomes a critical issue.
For some examples of good Residuals vs Fitted plot and ones that show serious errors, check out this page. Looking at these examples, you can see the first case has a plot in which the red line sticks pretty closely to the zero line, while the other cases show some serious deviation. Our plot for the lab is much closer to the ‘good’ plot than it is to the ‘serious issues’ plots. So we’ll consider our data okay and proceed with our analysis. Obviously, this is quite a subjective decision. The key takeaway is that these evaluations are closely tied to the context of our sample, our data, and what we’re studying. It’s almost always a judgement call.
You’ll notice in the bottom right corner, there are some points with numbers included: these are participants (“cases”, indicated by row number) who have the most influence on the regression line (and so they might be outliers). We’ll cover more about outliers in the next section.
[NOTE: All of the above text is informational. You do NOT need to edit it for the HW.]
plot(reg_model, 1) #Residual vs Fitted plot
Interpretation: Our Residual vs Fitted plot shows a slight curve, suggesting some minor non-linearity between our independent and dependent variables. But, we are okay to proceed with the regression.
6.4 Check for outliers
The plot below addresses leverage, or how much each data point is able to influence the regression line. Outliers are points that have undue influence on the regression line, the way that Bill Gates entering the room has an undue influence on the mean income.
The Cook’s distance plot is a visualization of a score called (you guessed it) Cook’s distance, calculated for each case (aka participant) in the dataframe. Cook’s distance tells us how much the regression would change if that data point was removed. Ideally, we want all points to have the same influence on the regression line, although we accept that there will be some variability. The cutoff for a high Cook’s distance score is .50. For our lab data, some points do exert more influence than others, but none of them are close to the cutoff. Remember, the plot will always identify the 3 most extreme values; it is your job to identify if any of those values are beyond the cutoff value.
[NOTE: All of the above text is informational. You do NOT need to edit it for the HW.]
# Cook's distance
plot(reg_model, 4)
Interpretation: Our data does not have any severe outliers.
6.5 Issues with My Data
Before interpreting our results, we checked whether our variables met the assumptions for simple linear regression. The Residuals vs Fitted plot showed slight non-linearity, but not enough to violate linearity. We also examined the Cook’s distance plot to check for outliers. All observations were well below the common cutoff of 1, so there was no influential outliers.
7 View Statistical Test Output
summary(reg_model)##
## Call:
## lm(formula = iou ~ big5_neu_std, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.8582 -0.4825 -0.0346 0.4586 3.3283
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.61217 0.01909 136.85 <2e-16 ***
## big5_neu_std 0.60643 0.01909 31.76 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.695 on 1324 degrees of freedom
## Multiple R-squared: 0.4324, Adjusted R-squared: 0.432
## F-statistic: 1009 on 1 and 1324 DF, p-value: < 2.2e-16# NOTE: For the write-up section below, to type lowercase Beta (ß) you need to hold down Alt key and type 225 on numeric keypad. If that doesn't work (upon releasing the Alt key), you should be able to copy/paste it from somewhere else in the write-up.Effect size, based on Regression ß (Beta Estimate) value in our output * Trivial: Less than 0.10 (ß < 0.10) * Small: 0.10–0.29 (0.10 < ß < 0.29) * Medium: 0.30–0.49 (0.30 < ß < 0.49) * Large: 0.50 or greater (ß > 0.50)
8 Write Up Results
To test our hypothesis that neuroticism significantly predicts intolerance of uncertainty, and that the relationship would be positive, we conducted a simple linear regression. Before interpreting results, we confirmed that our data met the key assumptions for linear regression. The Residuals vs Fitted plot showed slight non-linearity, but not enough to violate the assumption of linearity. The Cook’s distance plot indicated that no influential outliers were present. (Note: We are skipping the assumptions of normality and homogeneity of variance for this analysis.)
As predicted, we found that neuroticism significantly predicted intolerance of uncertainty, Adj. R2 = 0.43, F(1, 1324) = 1009, p < .001 . Additionally, the relationship between neuroticism and intolerance of uncertainity was positive, ß = 0.61, t() = 31.76 , p < .001 (refer to Figure 1). According to Cohen (1988), this constitutes a large effect size (ß > 0.50).
References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.