8.6: MLR inference - Parameter inferences using the t-distribution

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I have been deliberately vague about what an important variable is up to this point, and chose to focus on some bigger modeling issues. Now we turn our attention to one of the most common tasks in any basic statistical model – assessing whether a particular observed result is more unusual than we would expect by chance if it really wasn’t related to the response. The previous discussions of estimation in MLR models informs our interpretations of of the tests. The \(t\)-tests for slope coefficients are based on our standard recipe – take the estimate, divide it by its standard error and then, assuming the statistic follows a \(t\)-distribution under the null hypothesis, find a p-value. This tests whether each true slope coefficient, \(\beta_k\), is 0 or not, in a model that contains the other variables. Again, sometimes we say “after adjusting for” the other \(x\text{'s}\) or “conditional on” the other \(x\text{'s}\) in the model or “after allowing for”… as in the slope coefficient interpretations above. The main point is that you should not interpret anything related to slope coefficients in MLR without referencing the other variables that are in the model! The tests for the slope coefficients assess \(\boldsymbol{H_0:\beta_k = 0}\), which in words is a test that there is no linear relationship between explanatory variable \(k\) and the response variable, \(y\), in the population, given the other variables in model. The typical alternative hypothesis is \(\boldsymbol{H_0:\beta_k\ne 0}\). In words, the alternative hypothesis is that there is some linear relationship between explanatory variable \(k\) and the response variable, \(y\), in the population, given the other variables in the model. It is also possible to test for positive or negative slopes in the alternative, but this is rarely the first concern, especially when MLR slopes can occasionally come out in unexpected directions.

The test statistic for these hypotheses is \(\boldsymbol{t = \dfrac{b_k}{\textbf{SE}_k}}\) and, if our assumptions hold, follows a \(t\)-distribution with \(n-K-1\) df where \(K\) is the number of predictor variables in the model. We perform the test for each slope coefficient, but the test is conditional on the other variables in the model – the order the variables are fit in does not change \(t\)-test results. For the Snow Depth example with Elevation and Maximum Temperature as predictors, the pertinent output is in the four columns of the Coefficient table that is the first part of the model summary we’ve been working with. You can find the estimated slope (Estimate column), the SE of the slopes (Std. Error column), the \(t\)-statistics (t value column), and the p-values (Pr(>|t|) column). The degrees of freedom for the \(t\)-distributions show up below the coefficients and the \(df = 20\) here. This is because \(n = 23\) and \(K = 2\), so \(df = 23-2-1 = 20\).

m5 <- lm(Snow.Depth ~ Elevation + Max.Temp, data = snotel_s %>% slice(-c(9,22)))
summary(m5)

## 
## Call:
## lm(formula = Snow.Depth ~ Elevation + Max.Temp, data = snotel_s %>% 
##     slice(-c(9, 22)))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -14.652  -4.645   0.518   3.744  20.550 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.675e+02  3.924e+01  -4.269 0.000375
## Elevation    2.407e-02  3.162e-03   7.613 2.48e-07
## Max.Temp     1.253e+00  5.385e-01   2.327 0.030556
## 
## Residual standard error: 8.726 on 20 degrees of freedom
## Multiple R-squared:  0.8495, Adjusted R-squared:  0.8344 
## F-statistic: 56.43 on 2 and 20 DF,  p-value: 5.979e-09

The hypotheses for the Maximum Temperature term (Max.Temp) are:

\(\boldsymbol{H_0: \beta_{\textbf{Max.Temp}} = 0}\) given that Elevation is in the model vs
\(\boldsymbol{H_A: \beta_{\textbf{Max.Temp}}\ne 0}\) given that Elevation is in the model.

The test statistic is \(t = 2.327\) with \(df = 20\) (so under the null hypothesis the test statistic follows a \(t_{20}\)-distribution).

The output provides a p-value of \(0.0306\) for this test. We can also find this using pt:

2*pt(2.327, df = 20, lower.tail = F)

## [1] 0.03058319

The chance of observing a slope for Max.Temp as extreme or more extreme than assuming there really is no linear relationship between Max.Temp and Snow Depth (in a model with Elevation), is about 3% so this presents moderate evidence against the null hypothesis, in favor of retaining this term in the model.

Conclusion: There is moderate evidence against the null hypothesis of no linear relationship between Max.Temp and Snow Depth (\(t_{20} = 2.33\), p-value = 0.03), once we account for Elevation, so we can conclude that there likely is a linear relationship between them given Elevation in the population of SNOTEL sites in Montana on this day and we should retain this term in the model. Because we cannot randomly assign the temperatures to sites, we cannot conclude that temperature causes changes in the snow depth – in fact it might even be possible for a location to have different temperatures because of different snow depths. The inferences do pertain to the population of SNOTEL sites on this day because of the random sample from the population of sites.

Similarly, we can test for Elevation after controlling for the Maximum Temperature:

\[\boldsymbol{H_0: \beta_{\textbf{Elevation}} = 0 \textbf{ vs } H_A: \beta_{\textbf{Elevation}}\ne 0},\]

given that Max.Temp is in the model:

\(t = 7.613\) (\(df = 20\)) with a p-value of \(0.00000025\) or just \(<0.00001\).

So there is strong evidence against the null hypothesis of no linear relationship between Elevation and Snow Depth, once we adjust for Max.Temp in the population of SNOTEL sites in Montana on this day, so we would conclude that they are linearly related and that we should retain the Elevation predictor in the model with Max.Temp.

There is one last test that is of dubious interest in almost every situation – to test that the \(y\)-intercept \((\boldsymbol{\beta_0})\) in an MLR is 0. This tests if the true mean response is 0 when all the predictor variables are set to 0. I see researchers reporting this p-value frequently and it is possibly the most useless piece of information in the regression model summary. Sometimes less educated statistics users even think this result is proof of something interesting or are disappointed when the p-value is not small. Unless you want to do some prediction and are interested in whether the mean response when all the predictors are set to 0 is different from 0, this test should not be reported or, if reported, is certainly not very interesting¹⁴³. But we should at least go through the motions on this test once so you don’t make the same mistakes:

\(\boldsymbol{H_0: \beta_0 = 0 \textbf{ vs } H_A: \beta_0\ne 0}\) in a model with Elevation and Maximum Temperature.

\(t = -4.269\), with an assumption that the test statistic follows a \(t_{20}\)-distribution under the null hypothesis, and the p-value \(= 0.000375\).

There is strong evidence against the null hypothesis that the true mean Snow Depth is 0 when the Maximum Temperature is 0 and the Elevation is 0 in the population of SNOTEL sites, so we could conclude that the true mean Snow Depth is different from 0 at these values of the predictors. To reinforce the general uselessness of this test, think about the combination of \(x\text{'s}\) – is that even physically possible in Montana (or the continental US) in April?

Remember when testing slope coefficients in MLR, that if we find weak evidence against the null hypothesis, it does not mean that there is no relationship or even no linear relationship between the variables, but that there is insufficient evidence against the null hypothesis of no linear relationship once we account for the other variables in the model. If you do not find a small p-value for a variable, you should either be cautious when interpreting the coefficient, or not interpret it. Some model building strategies would lead to dropping the term from the model but sometimes we will have models to interpret that contain terms with larger p-values. Sometimes they are still of interest but the weight on the interpretation isn’t as heavy as if the term had a small p-value – you should remember that you can’t prove that coefficient is different from 0 in that model. It also may mean that you don’t know too much about its specific value. Confidence intervals will help us pin down where we think the true slope coefficient might be located, given the other variables in the model, and so are usually pretty interesting to report, regardless of how you approached model building and possible refinement.

Confidence intervals provide the dual uses of inferences for the location of the true slope and whether the true slope seems to be different from 0. The confidence intervals here have our regular format of estimate \(\mp\) margin of error. Like the previous tests, we work with \(t\)-distributions with \(n-K-1\) degrees of freedom. Specifically the 95% confidence interval for slope coefficient \(k\) is

\[\boldsymbol{b_k \mp t^*_{n-K-1}\textbf{SE}_{b_k}}\]

The interpretation is the same as in SLR with the additional tag of “after controlling for the other variables in the model” for the reasons discussed before. The general slope CI interpretation for predictor \(\boldsymbol{x_k}\) in an MLR is:

For a 1 [unit of \(\boldsymbol{x_k}\)] increase in \(\boldsymbol{x_k}\), we are 95% confident that the true mean of \(\boldsymbol{y}\) changes by between LL and UL [units of \(\boldsymbol{Y}\)] in the population, after adjusting for the other \(x\text{'s}\) [list them!].

We can either calculate these intervals as we have many times before or rely on the confint function to do this:

confint(m5)

##                     2.5 %       97.5 %
## (Intercept) -249.37903311 -85.67576239
## Elevation      0.01747878   0.03067123
## Max.Temp       0.13001718   2.37644112

So for a \(1^\circ F\) increase in Maximum Temperature, we are 95% confident that the true mean Snow Depth will change by between 0.13 and 2.38 inches in the population, after adjusting for the Elevation of the sites. Similarly, for a 1 foot increase in Elevation, we are 95% confident that the true mean Snow Depth will change by between 0.0175 and 0.0307 inches in the population, after adjusting for the Maximum Temperature of the sites.