5. Suppose you believe that a continuous response variable Y and a continuous pr

Question

5. Suppose you believe that a continuous response variable Y and a continuous predictor variable X are related as shown in the figure below. That is, Y increases linearly with X, but a change in slope may occur at a known change point Xu rO (a) How would you fit this model? State the model equation and explain. (b) How would you test the null hypothesis that the slopes over the two regions X S ro and X ro are equal? (c) How would you test the null hypothesis that E(YIX 5? Hint: When you write the equation of the model, you have a multiple linear regression model We have different tests for different hypotheses For example, for part (b), the slope will be the same, if you just have a simple linear model, that is, if one of the coefficients is zero. That is a t test or equivalently an F test. For part (e), remember the general F test

Explanation / Answer

a. We shall fit the partitioned type model in this case. As we can see that the slope changes. So an appropriate model will be

y ~ a + b1x *I(x > x0) + b2x * I(x<x0) + ...

... if there are other predictors or explanatory variables modelled usually.

We will estimate parameters by MLE and get the estimate of b1 and b2. For predicting, whenever we have x > x0, we have slope b1 and vice versa.

b. We have the estimates of b1 and b2. We can test b1-b2 = 0 as null hypothesis. If it is rejected, the slopes are different, otherwise slopes are same. We have wald's test and MLR test for that.

c. We have distribution of {hat{XB}} following normal with mean XB and variance depending on model. At given x, we can test for the mean of that by F-test.

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