EXST7034 - Regression Techniques Page 1POWER 1 in testing regression coefficients.œ"If we reject a null hypothesis, we need not concern ourselves with power. If wereject H , we have a (1- )100% chance of having made an error (called!!TYPE I error or error) in saying a difference exists (eg. 0) when in!""Áfact a difference did not exist. We set , and we know , if the assumptions!!are met.If we fail to reject the null hypothesis, we cannot make a TYPE I or error.!However, it is also possible that we have made an error in not demonstratinga difference exists when in fact one does exist. This error is called TYPE IIor error (not to be confused with regression coefficients “ ").""A error can be made only when H is true, and the probability of making this""error can be calculated only when we know the real difference between thehypothesized and true value of the regression coefficient. In practice thiscannot be known, so we never really know the probability of TYPE II error.For this reason, we never “accept" the null hypothesis, we only state that wecannot reject or that the null hypothesis is consistent with the data.However, if we are willing to “guess" at the difference between the hypothesizedand true values of the regression coefficients we can calculate a value for ,"the probability of a TYPE II error and POWER 1 .œ" Power P{| observed t | > tabular t | }œ" ß8#!#$where is called the , and is a standardized measure of$ noncentrality parameterhow far the value of the true parameter ( ) departs from the value of the""hypothesized parameter .""! = $()""5""!""In practice, is estimated by s and is estimated by b .5"bb""""However, b is not the true value of the parameter, only an estimate. All we can"state in calculating power is that “if b were the true value then the power"would be ...", or “if the difference between the true value and thehypothesized value is really ( ) then the power is ...".""""!EXST7034 - Regression Techniques Page 2Example of Power Calculation from Power Chart : Take a generic example. X Y 1 11 2 9, 10 3 17, 18, 8 4 7 5 16, 19 7 12Model Crossproducts X X X Y Y YwwwX X INTERCEP X YwINTERCEP 10 35 127X 35 151 465Y 127 465 1789X X Inverse, Parameter Estimates, and SSEw INTERCEP X YINTERCEP 0.5298245614 -0.122807018 10.18245614X -0.122807018 0.0350877193 0.7192982456Y 10.18245614 0.7192982456 161.35438596Dependent Variable: YAnalysis of Variance Sum of MeanSource DF Squares Square F Value Prob>FModel 1 14.74561 14.74561 0.731 0.4174Error 8 161.35439 20.16930C Total 9 176.10000Parameter Estimates Parameter Standard T for H0:Variable DF Estimate Error Parameter=0 Prob > |T|INTERCEP 1 10.182456 3.26897378 3.115 0.0143X 1 0.719298 0.84124591 0.855 0.4174Calculate power where is hypothesized to be 0. P = 0.4174."" $^ = = = 0.855 which is the t statistic(b )s 0.8412(0.7193 0)""!""b Now examine power table ( =0.05) with = 0.855 and 8 df.^!$ Power is only about 8%, or P(TYPE II error) = 1-0.08 = 0.92Calculate power where is hypothesized to be 0. P = 0.0143."! Now examine power table ( =0.05) with = 3.115 and 8 df.^!$ Power is only about 74%, or P(TYPE II error) = 1-0.74 =
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