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Suppose, I have performed multiple regression analysis on the following data set where X1 and X2 are independent variables and Y is the dependent variable.

Data Set

And achieved the following multiple regression analysis table from where it is clear that X2 is not correlated with Y as the p-value is greater than 0.05 which is 0.108 for X2 variable

Coefficient Table

So, if p value is greater than 0.05 for a variable, then why it is considered in regression equation

Y = 2.709 + 0.763(X1) + 0.463(X2) which has been shown in any online multiple regression equation calculator.

My question is Why it is not like the following Y = 2.709 + 0.763(X1) omitting the [0.463(X2)] part from the equation, as X2 is not correlated with Y

Please help to clarify the conceptual idea....

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  • $\begingroup$ if p value is greater than 0.05 for a variable, then why it is considered in regression equation ? It is a mythical manner, to conclude a positive or negative relationship between two postulates. $\endgroup$ – Subhash C. Davar Apr 28 '18 at 5:46
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One reason to keep $X_2$ in the model is (lack of) power:

The non-rejection implied by $p>0.05$ may be a type-II error, i.e., a wrong null hypothesis of irrelevance of $X_2$ that has wrongly not been rejected. Especially in situations like yours where the sample size is small, such problems may occur often.

Put somewhat differently, while the effect of $X_2$ is not statistically significant, it may still be significant from a subject matter point of view - 0.463 may be a "large" effect which we just may not be able to statistically distinguish from zero given the estimation uncertainty.

Also, note that, in any case, it would not be sound to report $Y = 2.709 + 0.763X_1$ as your fitted model. If you drop $X_2$ and reestimate your model with only $X_1$ as a regressor, the coefficient estimates will differ.

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The logic cited by you for inclusion (or exclusion) of a variablle as an explanatory variable in a regression imodel is ncorrect. The p- value =.108 indicates that in less than 10.8% of cases, t statistics is likely to exceed the observed statistics under the null reference distribution. The criterion of .05 can be surpassed if you want to allow more uncertainity (and lower validity of the results of your study).Thus theoretically; incorporating a variable in your model with a lower precision is not debarred. Moreover, you must know that whether there is a significant contribution by the added variable, could be knowm only after executing the regression model.

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