Prof. Maria Tackett
Oct 31, 2022
Project proposal due Fri, Nov 04
HW 03 due Mon, Nov 07 at 11:59pm
See Week 10 activities
Conditions for inference
Multicollinearity
rail_trail
# A tibble: 90 × 7
volume hightemp avgtemp season cloudcover precip day_type
<dbl> <dbl> <dbl> <chr> <dbl> <dbl> <chr>
1 501 83 66.5 Summer 7.60 0 Weekday
2 419 73 61 Summer 6.30 0.290 Weekday
3 397 74 63 Spring 7.5 0.320 Weekday
4 385 95 78 Summer 2.60 0 Weekend
5 200 44 48 Spring 10 0.140 Weekday
6 375 69 61.5 Spring 6.60 0.0200 Weekday
7 417 66 52.5 Spring 2.40 0 Weekday
8 629 66 52 Spring 0 0 Weekend
9 533 80 67.5 Summer 3.80 0 Weekend
10 547 79 62 Summer 4.10 0 Weekday
# … with 80 more rows
Source: Pioneer Valley Planning Commission via the mosaicData package.
Outcome:
volume
estimated number of trail users that day (number of breaks recorded)
Predictors
hightemp
daily high temperature (in degrees Fahrenheit)avgtemp
average of daily low and daily high temperature (in degrees Fahrenheit)season
one of “Fall”, “Spring”, or “Summer”cloudcover
measure of cloud cover (in oktas)precip
measure of precipitation (in inches)day_type
one of “weekday” or “weekend”Including all available predictors
Fit:
Summarize:
# A tibble: 8 × 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 17.6 76.6 0.230 0.819
2 hightemp 7.07 2.42 2.92 0.00450
3 avgtemp -2.04 3.14 -0.648 0.519
4 seasonSpring 35.9 33.0 1.09 0.280
5 seasonSummer 24.2 52.8 0.457 0.649
6 cloudcover -7.25 3.84 -1.89 0.0627
7 precip -95.7 42.6 -2.25 0.0273
8 day_typeWeekend 35.9 22.4 1.60 0.113
Linearity: There is a linear relationship between the response and predictor variables.
Constant Variance: The variability about the least squares line is generally constant.
Normality: The distribution of the residuals is approximately normal.
Independence: The residuals are independent from each other.
Does the linearity condition appear to be met?
Look at individual plots of residuals vs. each predictor, particularly if there are potential issues in the plot of residuals vs. predicted values.
The plots of residuals vs. hightemp
and avgtemp
appear to have a parabolic pattern.
The linearity condition is not satisfied given these plots.
Does the constant variance condition appear to be satisfied?
The vertical spread of the residuals is not constant across the plot.
The constant variance condition is not satisfied.
Ex 2. Fill in the code to recreate the following visualization in R based on the results of the model.
We can often check the independence condition based on the context of the data and how the observations were collected.
If the data were collected in a particular order, examine a scatterplot of the residuals versus order in which the data were collected.
If there is a grouping variable lurking in the background, check the residuals based on that grouping variable.
Residuals vs. order of data collection:
Residuals vs. predicted values by season
:
Residuals vs. predicted values by day_type
:
No clear pattern in the residuals vs. order of data collection plot and the model predicts similarly for seasons and day types.
Independence condition appears to be satisfied, as far as we can evaluate it.
We can’t include two variables that have a perfect linear association with each other
If we did so, we could not find unique estimates for the model coefficients
Suppose the true population regression equation is \(y = 3 + 4x\)
\[ \begin{aligned}\hat{y}&= \hat{\beta}_0 + \hat{\beta}_1x + \hat{\beta}_2z\\ &= \hat{\beta}_0 + \hat{\beta}_1x + \hat{\beta}_2\frac{x}{10}\\ &= \hat{\beta}_0 + \bigg(\hat{\beta}_1 + \frac{\hat{\beta}_2}{10}\bigg)x \end{aligned} \]
\[\hat{y} = \hat{\beta}_0 + \bigg(\hat{\beta}_1 + \frac{\hat{\beta}_2}{10}\bigg)x\]
We can set \(\hat{\beta}_1\) and \(\hat{\beta}_2\) to any two numbers such that \(\hat{\beta}_1 + \frac{\hat{\beta}_2}{10} = 4\)
Therefore, we are unable to choose the “best” combination of \(\hat{\beta}_1\) and \(\hat{\beta}_2\)
When we have almost perfect collinearities (i.e. highly correlated predictor variables), the standard errors for our regression coefficients inflate
In other words, we lose precision in our estimates of the regression coefficients
This impedes our ability to use the model for inference or prediction
Multicollinearity may occur when…
One (or more) predictor variables is an almost perfect linear combination of the others
Include a quadratic in the model mean-centering the variable first
Including interactions between two or more continuous variables
Variance Inflation Factor (VIF): Measure of multicollinearity in the regression model
\[VIF(\hat{\beta}_j) = \frac{1}{1-R^2_{X_j|X_{-j}}}\]
where \(R^2_{X_j|X_{-j}}\) is the proportion of variation \(X\) that is explained by the linear combination of the other explanatory variables in the model.
Typically \(VIF > 10\) indicates concerning multicollinearity - Variables with similar values of VIF are typically the ones correlated with each other
Use the vif()
function in the rms R package to calculate VIF
hightemp avgtemp seasonSpring seasonSummer cloudcover
10.259978 13.086175 2.751577 5.841985 1.587485
precip day_typeWeekend
1.295352 1.125741
hightemp
and avgtemp
are correlated. We need to remove one of these variables and refit the model.
hightemp
m1 <- linear_reg() |>
set_engine("lm") |>
fit(volume ~ . - hightemp, data = rail_trail)
m1 |>
tidy() |>
kable(digits = 3)
term | estimate | std.error | statistic | p.value |
---|---|---|---|---|
(Intercept) | 76.071 | 77.204 | 0.985 | 0.327 |
avgtemp | 6.003 | 1.583 | 3.792 | 0.000 |
seasonSpring | 34.555 | 34.454 | 1.003 | 0.319 |
seasonSummer | 13.531 | 55.024 | 0.246 | 0.806 |
cloudcover | -12.807 | 3.488 | -3.672 | 0.000 |
precip | -110.736 | 44.137 | -2.509 | 0.014 |
day_typeWeekend | 48.420 | 22.993 | 2.106 | 0.038 |
avgtemp
m2 <- linear_reg() |>
set_engine("lm") |>
fit(volume ~ . - avgtemp, data = rail_trail)
m2 |>
tidy() |>
kable(digits = 3)
term | estimate | std.error | statistic | p.value |
---|---|---|---|---|
(Intercept) | 8.421 | 74.992 | 0.112 | 0.911 |
hightemp | 5.696 | 1.164 | 4.895 | 0.000 |
seasonSpring | 31.239 | 32.082 | 0.974 | 0.333 |
seasonSummer | 9.424 | 47.504 | 0.198 | 0.843 |
cloudcover | -8.353 | 3.435 | -2.431 | 0.017 |
precip | -98.904 | 42.137 | -2.347 | 0.021 |
day_typeWeekend | 37.062 | 22.280 | 1.663 | 0.100 |
Model with hightemp removed:
adj.r.squared | AIC | BIC |
---|---|---|
0.42 | 1087.5 | 1107.5 |
Model with avgtemp removed:
adj.r.squared | AIC | BIC |
---|---|---|
0.47 | 1079.05 | 1099.05 |
Based on Adjusted \(R^2\), AIC, and BIC, the model with avgtemp removed is a better fit. Therefore, we choose to remove avgtemp from the model and leave hightemp in the model to deal with the multicollinearity.
term | estimate | std.error | statistic | p.value |
---|---|---|---|---|
(Intercept) | 8.421 | 74.992 | 0.112 | 0.911 |
hightemp | 5.696 | 1.164 | 4.895 | 0.000 |
seasonSpring | 31.239 | 32.082 | 0.974 | 0.333 |
seasonSummer | 9.424 | 47.504 | 0.198 | 0.843 |
cloudcover | -8.353 | 3.435 | -2.431 | 0.017 |
precip | -98.904 | 42.137 | -2.347 | 0.021 |
day_typeWeekend | 37.062 | 22.280 | 1.663 | 0.100 |
Conditions for inference
Multicollinearity