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Assuming that the desired significance is 0.1, since value is rejected and it can be concluded that is significant. The test for can be carried out in a similar manner. In the results obtained from the DOE folio, the calculations for this test are displayed in the ANOVA table as shown in the following figure. Note that the conclusion obtained in this example can also be obtained using the test as explained in the in . The ANOVA and Regression Information tables in the DOE folio represent two different ways to test for the significance of the variables included in the multiple linear regression model.

Here are three experiments to illustrate when the different approaches to statistics are appropriate. In the first experiment, you are testing a plant extract on rabbits to see if it will lower their blood pressure. You already know that the plant extract is a diuretic (makes the rabbits pee more) and you already know that diuretics tend to lower blood pressure, so you think there's a good chance it will work. If it does work, you'll do more low-cost animal tests on it before you do expensive, potentially risky human trials. Your prior expectation is that the null hypothesis (that the plant extract has no effect) has a good chance of being false, and the cost of a false positive is fairly low. So you should do frequentist hypothesis testing, with a significance level of 0.05.

In the simple linear regression model the true error terms, , are never known. The residuals, , may be thought of as the observed error terms that are similar to the true error terms. Since the true error terms, , are assumed to be normally distributed with a mean of zero and a variance of , in a good model the observed error terms (i.e., the residuals, ) should also follow these assumptions. Thus the residuals in the simple linear regression should be normally distributed with a mean of zero and a constant variance of . Residuals are usually plotted against the fitted values, , against the predictor variable values, , and against time or run-order sequence, in addition to the normal probability plot. Plots of residuals are used to check for the following:

The other values displayed with are S, R-sq(adj), PRESS and R-sq(pred). These values measure different aspects of the adequacy of the regression model. For example, the value of S is the square root of the error mean square, , and represents the "standard error of the model." A lower value of S indicates a better fitting model. The values of S, R-sq and R-sq(adj) indicate how well the model fits the observed data. The values of PRESS and R-sq(pred) are indicators of how well the regression model predicts new observations. R-sq(adj), PRESS and R-sq(pred) are explained in .

Examples of residual plots are shown in the following figure. (a) is a satisfactory plot with the residuals falling in a horizontal band with no systematic pattern. Such a plot indicates an appropriate regression model. (b) shows residuals falling in a funnel shape. Such a plot indicates increase in variance of residuals and the assumption of constant variance is violated here. Transformation on may be helpful in this case (see ). If the residuals follow the pattern of (c) or (d), then this is an indication that the linear regression model is not adequate. Addition of higher order terms to the regression model or transformation on or may be required in such cases. A plot of residuals may also show a pattern as seen in (e), indicating that the residuals increase (or decrease) as the run order sequence or time progresses. This may be due to factors such as operator-learning or instrument-creep and should be investigated further.

Conclusively, the multiple regression reveals that prestige rating is expected to be increasing as the level of education and income increases. In other words, prestige rating has a positve linear relationship with the predictors, education and income.

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The Hypothesis Testing About Coefficients of a Regression Model can be discussed in two themes. First, we will demonstrate how to test a single coefficient while the second case will explain hypothesis testing about multiple regression coefficients. Lets begin the first case, then.

We can see in the above output that the regression coefficients for all IVs are given with t and p-values. The hypothesis testing following like:

The null hypothesis is a statement that you want to test. In general, the null hypothesis is that things are the same as each other, or the same as a theoretical expectation. For example, if you measure the size of the feet of male and female chickens, the null hypothesis could be that the average foot size in male chickens is the same as the average foot size in female chickens. If you count the number of male and female chickens born to a set of hens, the null hypothesis could be that the ratio of males to females is equal to a theoretical expectation of a 1:1 ratio.

I have two regressor and one dependent variable. when I run multiple regression then ANOVA table show F value is 2.179, this mean research will fail to reject the null hypothesis. what should I do now, please give me some suggestions

For any of the variables _{j} included in a multiple regression model, the null hypothesis states that the coefficient _{j} is equal to 0.

From the ANOVA table output the p-value of 0.000 shows that we would reject the null hypothesis that all the slopes equal 0 and conclude that at least one of the slopes differs significantly from zero. , this does **not** tells us how many differ and/or which one(s) differ.

The following sections discuss hypothesis tests on the regression coefficients in simple linear regression. These tests can be carried out if it can be assumed that the random error term, , is normally and independently distributed with a mean of zero and variance of .

There are different ways of doing statistics. The technique used by the vast majority of biologists, and the technique that most of this handbook describes, is sometimes called "frequentist" or "classical" statistics. It involves testing a null hypothesis by comparing the data you observe in your experiment with the predictions of a null hypothesis. You estimate what the probability would be of obtaining the observed results, or something more extreme, if the null hypothesis were true. If this estimated probability (the *P* value) is small enough (below the significance value), then you conclude that it is unlikely that the null hypothesis is true; you reject the null hypothesis and accept an alternative hypothesis.

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