Is used to test the hypothesis that the values of the regression parameters
Video TranscriptWell, it's of the regulation parameters. That is B one, b two dot dot dot vic. You are all zero. So the correct option for this question is F test. The correct option? Age F test ap test is a statistical test ap test each. Uh Is there a statistical test, which is the mhm null hypothesis of the regulation values better one better to do what are not we take. You are all, you know, the artist can be used in many models parallel. E hence the Great Arms Reach. Show
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. The research questions and their corresponding hypotheses are: Is the regression model containing at least one predictor useful in predicting the size of the infarct? Is the size of the infarct significantly
(linearly) related to the area of the region at risk? (Primary research question) Is the size of the infarct area significantly (linearly) related to the type of treatment upon controlling for the size of the region at risk for infarction? Let's test each of the hypotheses now using the general linear F-statistic: \(F^*=\left(\dfrac{SSE(R)-SSE(F)}{df_R-df_F}\right) \div \left(\dfrac{SSE(F)}{df_F}\right)\) 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. The number of error degrees of freedom associated with the reduced and full models —
\(df_{R}\) and \(df_{F}\), respectively — is the number of observations, n, minus the number of parameters, p, in the model. That is, in general, the number of error degrees of freedom is n-p. We use statistical software, such as Minitab's F-distribution probability calculator, to determine the P-value for each test. To answer the research question: "Is the regression model containing at least one predictor useful in predicting the size of the infarct?" To do so, we test the hypotheses: The full model is the largest possible model — that is, the model containing all of the possible predictors. In this case, the full model is: \(y_i=(\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i3})+\epsilon_i\) The error sum of squares for the full model, SSE(F), is just the usual error sum of squares, SSE, that appears in the analysis of variance table.
Because there are 4 parameters in the full model, the number of error degrees of freedom associated with the full model is \(df_{F} = n - 4\). The reduced model is the model that the null hypothesis describes. Because the null hypothesis sets each of the slope parameters in the full model equal to 0, the reduced model is: \(y_i=\beta_0+\epsilon_i\) The reduced model suggests that none of the variations in the response y is explained
by any of the predictors. Therefore, the error sum of squares for the reduced model, SSE(R), is just the total sum of squares, SSTO, that appears in the analysis of variance table. Because there is only one parameter in the reduced model, the number of error degrees of freedom associated with the reduced model is \(df_{R} = n - 1 \). Upon plugging in the above quantities, the general linear F-statistic: \(F^*=\dfrac{SSE(R)-SSE(F)}{df_R-df_F}
\div \dfrac{SSE(F)}{df_F}\) becomes the usual "overall F-test": \(F^*=\dfrac{SSR}{3} \div \dfrac{SSE}{n-4}=\dfrac{MSR}{MSE}\) That is, to test \(H_{0}\) : \(\beta_{1} = \beta_{2} = \beta_{3} = 0 \), we just use the overall F-test and P-value reported in the analysis of variance table: Inf = - 0.135 + 0.613 Area - 0.2435 X2 - 0.0657 X3 There is sufficient evidence (F = 16.43, P < 0.001) to conclude that at least one of the slope parameters is not equal to 0. In general, to test that all of the slope parameters in a multiple linear regression model are 0, we use the overall F-test reported in the analysis of variance table. Testing one slope parameter is 0 Section
Now let's answer the second research question: "Is the size of the infarct significantly (linearly) related to the area of the region at risk?" To do so, we test the hypotheses:
The TestThe general linear statistic: \(F^*=\dfrac{SSE(R)-SSE(F)}{df_R-df_F} \div \dfrac{SSE(F)}{df_F}\) simplifies to: \(F^*=\dfrac{SSR(x_1|x_2, x_3)}{1}\div \dfrac{SSE(x_1,x_2, x_3)}{n-4}=\dfrac{MSR(x_1|x_2, x_3)}{MSE(x_1,x_2, x_3)}\) Getting the numbers from the Minitab output: Analysis of Variance
Regression EquationInf = - 0.135 + 0.613 Area - 0.2435 X2 - 0.0657 X3 we determine that the value of the F-statistic is:
Testing a subset of slope parameters is 0 SectionFinally, let's answer the third — and primary — research question: "Is the size of the infarct area significantly (linearly) related to the type of treatment upon controlling for the size of the region at risk for infarction?" To do so, we test the hypotheses:
Summary of MLR Testing SectionFor the simple linear regression model, there is only one slope parameter about which one can perform hypothesis tests. For the multiple linear regression model, there are three different hypothesis tests for slopes that one could conduct. They are:
We have learned how to perform each of the above three hypothesis tests. Along the way, we also took two detours — one to learn about the "general linear F-test" and one to learn about "sequential sums of squares." As you now know, knowledge about both is necessary for performing the three hypothesis tests. The F-statistic and associated p-value in the ANOVA table is used for testing whether all of the slope parameters are 0. 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. For example, for the rabbit heart attacks study, the F-statistic is (0.95927/(4–1)) / (0.54491/(32–4)) = 16.43 with p-value 0.000. To test whether a subset — more than one, but not all — of the slope parameters are 0, there are two equivalent ways to calculate the F-statistic:
For example, for the rabbit heart attacks study, the general linear F-statistic is ((0.8793 – 0.54491) / (30 – 28)) / (0.54491 / 28) = 8.59 with p-value 0.0012. Alternatively, the partial F-statistic for testing the slope parameters for predictors \(x_{2}\) and \(x_{3}\) using sequential sums of squares is ((0.31453 + 0.01981) / 2) / (0.54491 / 28) = 8.59. To test whether one slope parameter is 0, we can use an F-test as just described. 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 example, for the rabbit heart attacks study, the F-statistic for testing the slope parameter for the Area predictor is (0.63742/1) / (0.54491/(32–4)) = 32.75 with p-value 0.000. Alternatively, the t-statistic for testing the slope parameter for the Area predictor is 0.613 / 0.107 = 5.72 with p-value 0.000, and \(5.72^{2} = 32.72\). Incidentally, you may be wondering why we can't just do a series of individual t-tests to test whether a subset of the slope parameters is 0. For example, for the rabbit heart attacks study, we could have done the following:
The problem with this approach is we're using two individual t-tests instead of one F-test, which means our chance of drawing an incorrect conclusion in our testing procedure is higher. Every time we do a hypothesis test, we can draw an incorrect conclusion by:
Thus, in general, the fewer tests we perform the better. In this case, this means that wherever possible using one F-test in place of multiple individual t-tests is preferable. Try it! Hypothesis tests for the slope parameters SectionThe problems in this section are designed to review the hypothesis tests for the slope parameters, as well as to give you some practice on models with a three-group qualitative variable (which we'll cover in more detail in Lesson 8). We consider tests for:
(Note the correct specification of the alternative hypotheses for the last two situations.) Sugar beets studyA group of researchers was interested in studying the effects of three different growth regulators (treat, denoted 1, 2, and 3) on the yield of sugar beets (y = yield, in pounds). They planned to plant the beets in 30 different plots and then randomly treat 10 plots with the first growth regulator, 10 plots with the second growth regulator, and 10 plots with the third growth regulator. One problem, though, is that the amount of available nitrogen in the 30 different plots varies naturally, thereby giving a potentially unfair advantage to plots with higher levels of available nitrogen. Therefore, the researchers also measured and recorded the available nitrogen (\(x_{1}\) = nit, in pounds/acre) in each plot. They are interested in comparing the mean yields of sugar beets subjected to the different growth regulators after taking into account the available nitrogen. The Sugar Beets dataset contains the data from the researcher's experiment.
What is used to test the hypothesis that the values of the regression parameters are all zero?A one-sample t-test will be used in linear regression to test the null hypothesis that the slope or the coefficient is equal to zero. In the case of the multiple regression model, the null hypothesis is that the coefficient of each of the predictor variables is equal to zero.
How do you test the hypothesis in a regression analysis?Hypothesis testing is used to confirm if our beta coefficients are significant in a linear regression model.. Formulate a Hypothesis.. Determine the significance level.. Determine the type of test.. Calculate the Test Statistic values and the p values.. Make Decision.. Which hypothesis is used in regression?In the case of the linear regression model, two types of hypothesis testing are done. They are T-tests and F-tests. In other words, there are two types of statistics that are used to assess whether linear regression models exist representing response and predictor variables. They are t-statistics and f-statistics.
Is used to test the hypothesis that the values of the regression parameters B0 B1 B2 BQ are all zero?what is used to test the hypothesis that the values of the regression parameters B0, B1, B2, ... Bq are all zero. multicollinearity.
|