Decision Making Using ANOVA in Regression | Applied Biostatistics | BIO733_Topic181

Duration: 7:00 944 transcript words EN summary

Video summary

In this module, the focus is on utilizing SPSS to perform Analysis of Variance (ANOVA) specifically for making decisions regarding a simple linear regression model. This statistical approach examines the relationship between one independent variable and one dependent variable, defined by an intercept ($\beta_0$), a slope ($\beta_1$), and a random error term. The primary objective is to determine whether there is a significant linear relationship between the variables in the population from which the sample was drawn. This is achieved through hypothesis testing where the null hypothesis states that the regression coefficients are equal to zero, implying no linear relationship, while the alternative hypothesis suggests that at least one coefficient is not zero. Although simple linear regression technically tests only the slope, the vector notation is often used to maintain consistency with multiple regression models. To illustrate this process, the video presents a clinical example involving Glomerular Filtration Rate (GFR) and inverse state C values in renal transplant recipients. Using data from 27 tests where DTPA GFR serves as the predictor and inverse state C as the response variable, an ANOVA table is constructed to analyze the sources of variation: total, regression, and residual. The degrees of freedom for the regression source are calculated by subtracting one from the number of parameters in the model (which is two), resulting in one degree of freedom. The remaining degrees of freedom, calculated as $n - k$, are assigned to the residual or error term. By dividing the mean square of the regression by the mean square of the residual, an F-statistic is derived, which follows an F-distribution under the assumption that the null hypothesis is true and model assumptions are met. The final steps involve interpreting the output generated by SPSS to make a statistical decision. In the provided example, the calculated F-ratio is approximately 35.177, leading to a p-value that is effectively zero, well below the significance level of 0.05. Since the p-value is less than alpha, the null hypothesis is rejected in favor of the alternative hypothesis. This rejection allows for the confident conclusion that there is indeed a statistically significant linear relationship between the independent variable (DTPA GFR) and the dependent variable (inverse state C). Ultimately, this analysis confirms that the regression model provides a valid explanation for the variation in the data, supporting the use of DTPA GFR as a reliable predictor for inverse state C.

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In this module, we'll learn to obtain ANOVA using SPSS. Along with that, we will also use this to make the decision about the regression model. Here we are talking about the simple linear regression model with one independent variable and one dependent variable with beta KN as the y intercept and beta 1 as a slope. and the random error term. So we speak of this model as regression Y on X. Here we wish to know if we can conclude that in the population from which our sample was drawn X and Y are linearly related. And to discuss this question, we carry out a test of hypothesis where hnot is beta vector is equals to zero against alternative beta vector is not equals to zero. Though in the case of simple linear regression model, one can simply write beta 1 is equals to 0 for null hypothesis and beta 1 not equals to 0 for alternative hypothesis. But uh I intentionally put on beta vector is equals to 0 and beta vector is not equals to zero as the hypothesis because this is going to be used similar way if we are using multiple linear regression model here if h not is rejected then we should be able to conclude that x and y are linearly related. Now to test this hypothesis, let's consider this example where we have GFR which the most important parameter of renal function assessed in renal transplant recipients. All the inulin clearance is regarded as a gold standard measure of GFR. Its use is in clinical practice is limited. Chrysler at all examined the relationship between the inverse of sustainance C and inulin GFR as measured by DTPA GFR clearance. So the results of 27 tests are shown in the following table and here again we are using DTPA GFR as a predictor of inverse state and C as the response variable. A data is given where DTPA, GFR and inverse of state C values are given assuming that the simple linear regression model and its underlying assumption are applicable. We want to test the hypothesis H not beta is equals to Z against alternative that beta not equals to 0 at 5% level of significance. Our ANOVA table will look like as given in this table here with source of variation as total linear regression and residual three sources of variation with their respective sum of squares degrees of freedom mean squares and variance ratio. So from the three sum of square terms and their associated degrees of freedom the analysis of variance table may be constructed and in general the degrees of freedom associated with the sum of squares due to regression is equals to the number of constants in the regression equation minus one. But in the case of a simple linear regression model we have two estimates beta KN and beta 1. Hence the degrees of freedom for regression are 2 minus one which is 1. It's very important that we understand what's the distribution of the test statistics and it can be shown that when the hypothesis of no linear relationship between X and Y is true and when assumption underlying regression are met the ratio obtained by dividing the regression mean square by the residual mean square is distributed as F with 1 and n minus2 degrees of freedom. And our decision rule states that we reject Hnot if the computed value of variance ratio is equal to or greater than the critical value of f. And we can also use the p value approach where we reject the null hypothesis if p value is less than alpha. Once a data is added to SPSS to see the ANOVA and regression we'll go to analyze regression and linear here our dependent variable and our independent variable will go to their respective categories and we keep the method to be enter and press okay. It gives us the output with four tables. The first table is for variable which are entered or removed from the model. Right now only GFR as an independent variable has been added. Having one independent variable means it's a simple linear regression model with the dependent variables state C inverse with the results for the model summary and the ANOVA table. Here we can observe there are three sources of variation here in our model. It's a total, its regression and its residual which is also called error with their respective sum of squares and degrees of freedom. Since we have 27 observations in our data, therefore the degrees of freedom for total is 27 - 1 which is 26. We have only two parameters in our regression model beta KN and beta 1. Hence our regression degrees of freedom is 2 minus one and remaining degree of freedom that is n minus k is 25 which is for the residual with their respective mean square and if you divide this 0.295 by 1 you'll get this answer and divide 0.210 2110 by 25 you'll get this answer. Taking this ratio the F ratio which is the variance ratio is 35.177 and using this F statistics the P value calculated is 0 which is very less than 0.001 001 and we can use this p value to make our decision that since this p value is less than alpha which is 0.05 05. Hence we may reject the null hypothesis beta is equals to 0. And we say that the alternative hypothesis is true. And when we say the alternative hypothesis is true, we can confidently say that there is X and Y are linearly related.