For selection of independent variables in multiple regression analysis, first have calculated correlation coefficient between dependent (y) and independent variables (x1, x2, x3, x4 & x5) and then I performed multiple regression analysis using significantly correlated variables. Result shows:
y & x1 = 0.91; y & x2 = 0.90; y & x3 = 0.23; y & x4 = 0.18; y & x5 = 0.23; all the correlation values are significant (pval <0.05).
In multiple regression to derive linear model, I started with two predictors, x1 & x2 which are highly correlated and sequentially added the rest of the predictors.
For model1 (x1 & x2) the R-square value is 0.88 and coefficient values of x1 and x2 are significant.
For model2 (x1, x2 & x3) the R-square value is 0.88 and coefficient values are significant for x1 & x2 while insignificant for x3.
Similar R-square value is obtained for model3 (x1, x2, x3 & x4) and the coefficent values are insignificant for x3 & x4.
For model4 (x1, x2, x3, x4 & x5) the higher R-square value (0.90) is obtained and coefficient values of x1 and x5 are significant while other variables (x2, x3 & x4) shows insignificant coefficient value.
My question is, in model4 why x2 is showing insignificant coefficient value while it is highly correlated with y? And finally which model I should consider for prediction of y?
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I made some changes in analysis. I removed x5 variable and added another variable instead of net radiation. So, this time I used x1 (rainfall), x2 (soil moisture), x3 (temperature) and x4 (vegetation data) and all models, passed the test of significance in the regression at a p value of 0.05.Now result shows: For model1 (x1 & x2) the R-square value is 0.88 and standard error is 18.29, For model2 (x1, x2 & x3) the R-square value is 0.89 and standard error is 17.53, For model3 (x1, x2, x3 & x4) R-square value is 0.89 and standard error is 17.42. You can see R-square and standard error values are almost similar. In this case, can I calculate accuracy (in percentage) in each model?