We can — finally — get back to the whole point of this lesson, namely learning how to conduct hypothesis tests for the slope parameters in a multiple regression model. What happens if we simultaneously add two predictors to a model containing only one predictor? How much did the error sum of squares decrease — or alternatively, the regression sum of squares increase? In general, the number appearing in each row of the table is the sequential sum of squares for the row’s variable given all the other variables that come before it in the table. When fitting a regression model, Minitab outputs Adjusted (Type III) sums of squares in the Anova table by default.
Student heights and GPAs
A sequential sum of squares quantifies how much variability we explain (increase in regression sum of squares) or alternatively how much error we reduce (reduction in the error sum of squares). In essence, when we add a predictor to a model, we hope to explain some of the variability in the response, and thereby reduce some of the error. The numerator of the general linear F-statistic — that is, \(SSE(R)-SSE(F)\) is what is referred to as a “sequential sum of squares” or “extra sum of squares.” Along the way, however, we have to take two asides — one to learn about the “general linear F-test” and one to learn about “sequential sums of squares.” Knowledge about both is necessary for performing the three hypothesis tests.
In this case, there appears to be no advantage in using the larger full model over the simpler reduced model. Let’s get a better feel for the general linear F-test approach by applying it to two different datasets. It doesn’t appear as if the reduced model would do a very good job of summarizing the trend in the population. What does the reduced model do for the skin cancer mortality example?
This concludes our discussion of our first aside from the general linear F-test. How different does SSE(R) have to be from SSE(F) in order to justify using the larger full model? Where are we going with this general linear test approach? That is, adding latitude to the model substantially reduces the variability in skin cancer mortality. That is, adding height to the model does very little in reducing the variability in grade point averages.
- The easiest way to learn about the general linear test is to first go back to what we know, namely the simple linear regression model.
- For example, suppose we have 3 predictors for our model.
- Now, we move on to our second aside from sequential sums of squares.
- Alternatively, we can use a t-test, which will have an identical p-value since in this case, the square of the t-statistic is equal to the F-statistic.
- For the multiple linear regression model, there are three different hypothesis tests for slopes that one could conduct.
Example 6-1: Heart attacks in Rabbits
Further, each predictor must have the same value for at least two observations for it to be considered a replicate. However, now we have p regression parameters and c unique X vectors. The Sugar Beets dataset contains the data from the researcher’s experiment. Alternatively, we can use a t-test, which will have an identical p-value since in this case, the square of the t-statistic is equal to the F-statistic. We have learned how to perform each of the above three hypothesis tests. We use statistical software, such as Minitab’s F-distribution probability calculator, to determine the P-value for each test.
To calculate the F-statistic for each test, we first determine the error sum of squares for the reduced and full models — SSE(R) and SSE(F), respectively. So far, we’ve only evaluated how much the error and regression sums of squares change when adding one additional predictor to the model. Perhaps, you noticed from the previous illustration that the order in which we add predictors to the model determines the sequential sums of squares (“Seq SS”) we get. For a given data set, the total sum of squares will always be the same regardless of the number of predictors in the model. The amount of error that remains upon fitting a multiple regression model naturally depends on which predictors are in the model.
Example 6-4: Peruvian Blood Pressure Data
What we need to do is to quantify how much error remains after fitting each of the two models to our data. How do we decide if the reduced model or the full model does a better job of describing the trend in the data when it can’t be determined by simply looking at a plot? The easiest way to learn about the general linear test is to first go back to what we know, namely the simple linear regression model. As you can see by the wording of the third step, the null hypothesis always pertains to the reduced model, while the alternative hypothesis always pertains to the full model.
extraF.nls: Compare Two \(nls\) Models Using Extra Sum-of-Squares F-Tests
To investigate their hypothesis, the researchers conducted an experiment on 32 anesthetized rabbits that were subjected to a heart attack. In this lesson, we learn how to perform each of the above three hypothesis tests.
6 – Lack of Fit Testing in the Multiple Regression Setting
That if all individuals had the same fit, this would not influence extra sum of squares). More general model (2), respectively. Different grouping levels in the dataset may be obscured when curves are fitted to the This function is not promoted for use in model selection as differences in curves of Check that models are nested prior to use. The function will produce seemingly adequate output with non-nested models.
Adjusted sums of squares measure the reduction in the error sum of squares (or increase in the regression sum of squares) when each predictor is added to a model that contains all of the remaining predictors. That is, the error sum of squares (SSE) and, hence, the regression sum of squares (SSR) depend on what predictors are in the model. The reduced model includes only the two variables LeftArm and LeftFoot as predictors.
Heart attacks in rabbits (revisited)
We first have to take two side trips — the first one to learn what is called “the general linear F-test.” Unfortunately, we can’t just jump right into the hypothesis tests. We’ll soon learn how to think about the t-test for a single slope parameter in the multiple regression framework. We’ll soon see that the null hypothesis is tested using the analysis of variance F-test. In this lesson, we learn how to perform three different hypothesis tests for slope parameters in order to answer various research questions.
If we obtain a large percentage, then it is likely we would want to specify some or all of the remaining predictors to be in the final model since they explain so much variation. In most applications, this p-value will be small enough to reject the null hypothesis and conclude that at least one predictor is useful in the model. At the beginning of this lesson, we translated three different research questions pertaining to heart attacks in rabbits (Cool Hearts dataset) into three sets of hypotheses we can test using the general linear F-statistic.
- Let’s revisit the Allen Cognitive Level Study data to see what happens when we reverse the order in which we enter the predictors in the model.
- So far, we’ve only evaluated how much the error and regression sums of squares change when adding one additional predictor to the model.
- The proportion of variation explained by the predictors in group B that cannot be explained by the predictors in group A is given by
- We also see that all four individual x-variables are statistically significant.
- Let’s try out the notation and the two alternative definitions of a sequential sum of squares on an example.
Testing one slope parameter is 0
First, we run a multiple regression using all nine x-variables as predictors. Formal lack of fit testing can also be performed in the multiple regression setting; however, the ability to achieve replicates can be more difficult as more predictors are added to the model. Let’s revisit the Allen Cognitive Level Study data to see what happens when we reverse the order in which we enter the predictors in the model. The total sum of squares quantifies how much the response varies — it has nothing to do with which predictors are in the model.
I’m hoping this example clearly illustrates the need for being able to “translate” a research question into a statistical procedure. Similarly, \(\beta_3\) represents the difference in the mean size of the infarcted area — controlling for the size of the region at risk —between “late cooling” and “no cooling” rabbits. Thus, \(\beta_2\) represents the difference in the mean size of the infarcted area — controlling for the size of the region at risk —between “early cooling” and “no cooling” rabbits.
For simple linear regression, it turns out that the general linear F-test is just the same ANOVA F-test that we learned before. In this case, there appears to be a big advantage in using the larger full model over the simpler reduced model. Here, there is quite a big difference between the estimated equation for the full model extrasum (solid line) and the estimated equation for the reduced model (dashed line).
The first calculation we will perform is for the general linear F-test. The Minitab output for the full model is given below. If this null is not rejected, it is reasonable to say that none of the five variables Height, Chin, Forearm, Calf, and Pulse contribute to the prediction/explanation of systolic blood pressure.
Hypotheses 2
The “full model”, which is also sometimes referred to as the “unrestricted model,” is the model thought to be most appropriate for the data. Once we understand the general linear test for the simple case, we then see that it can be easily extended to the multiple-case model. We will learn a general linear F-test for testing such a hypothesis. How could the researchers use the above regression model to answer their research question? Parameter model is to be preferred.