An F test for whether the simple linear regression model "explains" (really, predicts) a "significant" amount of the variance in the response. What this really does is compare two versions of the simple linear regression model. The null hypothesis is that all of the assumptions of that model hold, and the slope, B1 is exactly 0. This is sometimes called the "intercept-only" model, for obvious reasons. The alternative is that all of the simple linear regression assumptions hold with " B1 is the element of R"
The alternative, nin-zero- slope model will always fit the data better than the null, intercept- the only model; the Ftest asks if the improvement in fit is larger than we would expect under the null.
there are situations where it is useful to know about this precise quantity and so run an F test on the regression. It is hardly ever, however, a good way to check whether the simple linear regression model is correctly specified, because of neither retaining for rejecting the mull gives us information about what we really want to know.
Suppose first that we retain the null hypothesis, i.e., we do not find any significant share of variance associated with the regression. This could be because (i) the intercept-only model is right; (ii)B1 is not equal to 0 but the test does not have enough power to detect departures from the null. We do not know which it is There is also the possibility that the real relationship is nonlinear, but the best linear approximation to it has a slope (nearly) zero, in which case the F test will have no power to detect the nonlinearity.
suppose instead that we reject the null, intercept-the only hypothesis. This does not mean that the simple linear model is right. It means that the latter model predicts better than the intercept-only model - too much better to be due to chance. The simple linear regression model can be absolute garbage, with every single one of its assumptions flagrantly violated, and yet better than the model which makes all those assumptions and thinks the optimal slope is zero.
Neither the F test of B1=0 vs. B1 is not equal to 0 nor the Wald/t-test of the same hypothesis tell us anything about the correctness of the simple linear regression model. All these tests presume the simple linear regression model with Gaussian noise is true, and check a special case (flat line) against the general one (titled line). They do not test linearity, constant variance, lack of correlation. or Gaussianity.
The alternative, nin-zero- slope model will always fit the data better than the null, intercept- the only model; the Ftest asks if the improvement in fit is larger than we would expect under the null.
there are situations where it is useful to know about this precise quantity and so run an F test on the regression. It is hardly ever, however, a good way to check whether the simple linear regression model is correctly specified, because of neither retaining for rejecting the mull gives us information about what we really want to know.
Suppose first that we retain the null hypothesis, i.e., we do not find any significant share of variance associated with the regression. This could be because (i) the intercept-only model is right; (ii)B1 is not equal to 0 but the test does not have enough power to detect departures from the null. We do not know which it is There is also the possibility that the real relationship is nonlinear, but the best linear approximation to it has a slope (nearly) zero, in which case the F test will have no power to detect the nonlinearity.
suppose instead that we reject the null, intercept-the only hypothesis. This does not mean that the simple linear model is right. It means that the latter model predicts better than the intercept-only model - too much better to be due to chance. The simple linear regression model can be absolute garbage, with every single one of its assumptions flagrantly violated, and yet better than the model which makes all those assumptions and thinks the optimal slope is zero.
Neither the F test of B1=0 vs. B1 is not equal to 0 nor the Wald/t-test of the same hypothesis tell us anything about the correctness of the simple linear regression model. All these tests presume the simple linear regression model with Gaussian noise is true, and check a special case (flat line) against the general one (titled line). They do not test linearity, constant variance, lack of correlation. or Gaussianity.
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