# Global p value in multiple linear regression in R

I am using multiple linear regression to see association between length of hospital stay (Dependent Variable) and few predictors (Independent variables). All the predictors are categorical variables (factors). Some of the predictors have >2 subgroups. The Regression analysis gives p values for each sub group comparing with the reference subgroup. Is there a way to get a global p value for the predictor as a whole and not for each sub group?

• This is multiple regression, not multivariate regression. – whuber May 24 '19 at 15:51

Have you plotted the distribution of your length of hospital stay variable? I think you will likely find that the distribution is skewed to the right (since some patients will have much longer hospital stays compared to all other patients). This skewness will likely be inherited by the regression model residuals - if that is the case, you will need to address it if it is very pronounced (especially if your sample size is small). To address it, you may have to use median regression instead of ordinary least squares regression, or perhaps transform length of hospital stay before analysing it via ordinary least squares regression (e.g., you can try a log transformation or a square root transformation).

To get a global p-value for testing the effect of a categorical predictor in your ordinary least squares regression model, you can use a so-called partial F-test. This test allows you to compare the following two models:

Null model: The model which DOES NOT include the dummy variables corresponding to the categorical predictor whose effect on the length of hospital stay you want to test (adjusting for the effects of the other categorical predictors in your model)

Alternative model: The model which DOES include those dummy variables

Note that the null model is considered to be nested within the alternative model, since it is obtained from the alternative model by setting the coefficients of the dummy variables of interest to zero. The null model is therefore simpler than the alternative model; conversely, the alternative model is more complex than the simpler model. The partial F-test will test whether the more complex model is better than the simpler model. (See slides 15 and 16 of http://www.statpower.net/Content/311/Lecture%20Notes/RegressionIntro.pdf for the formula involved in defining the partial F-square statistic and its numerator and denominator degrees of freedom.)

The p-value of the partial F-test will thus help you test these two hypotheses:

Ho: The categorical predictor of interest has no effect on length of hospital stay, after adjusting for the effects of the other categorical predictors present in the model

Ha: The categorical predictor of interest has an effect on length of hospital stay, after adjusting for the effects of the other categorical predictors present in the model

Your software should be able to perform partial F-tests - you just have to read the documentation to find the corresponding details.

• Thanks for your reply. I have log transformed the Length of Hospital stay to achieve normal distribution and subsequently have run the multiple regression. Can i use Wald test (which gives a chi-squared test output) to test significance of the categorical variable. I have used it in another analysis involving logistic regression. I am not sure if i can use the same for linear regression. – CBGodbole May 30 '19 at 8:42

Deviance test (analysis of variance) is an option here, you can create two models to test the hypothesis of significance of including a predictor. The full model including the factor and a reduced model excluding the factor.

Then you can compare the log-likelihood and use deviance test (-2 * log-likelihood). By comparing the deviance of two models (using F statistic), you can know the significance of excluding this factor.

I believe what you're asking about is called MANOVA. This is just like ANOVA except the outcome is multivariate. One common test is based on Wilkes $$\lambda$$, and gives an overall test of whether the mean vector of the multivariate distribution varies between groups. The problem is that it doesn't provide specificity on which element(s) of the mean vector are different. For that you could use post-hoc tests, or perhaps univariate ANOVAs.

• I feel like MANOVA is for comparing two or more dependent variables rather than testing, where OP's situation only have 1 dependent variable. A simple ANOVA should just work here. – TYZ May 24 '19 at 15:48
• @YilunZhang, I was just reading the title :) Good point! – gammodel May 24 '19 at 18:23