I have posted this question before but wanted to give a more thorough explanation of my problem so as to hopefully garner some assistance. I am also posting it on Talk Stats under the same title.
I have a mutiple regression model with over 10,000 records and though the model has an R^2 of 91%, the lack of fit tests and slightly S-shaped normal probability plot of the residuals show suggest curvature in the data. My objective here is to figure out and correct/ transform the curvature so as to fit the fit appropriately.
I have tried every method I have researched that could possibly help, though it is possible that I still missed something. Here is what I have done:
1) Added quadratic and cubic terms of each x. This only seemed to exacerbate the curavture. I also tried only added these terms for one x at a time and the results were the same.
2) Used the natural log of y and all x's. This again exacerbated the S shaped residuals plot.
3) I have 3 categorical variables and thinking that variance between them was an issue, I added weights for each category by 1/each category's variance. This also yielded no benefit to the residual plot or fit of the model.
UPDATED PER MICHELLE'S REQUEST: --I am trying to predict sales volume by the amount to which our price is more or less than our main compeitor. I am in a high volume commoditized industry so I expect to be able to get some good results.
--My response is sales volume (y) and my predictors are the 4 week average of the sales volume for that same day and half-hour period (x1) and the amount we are above or below (in cents) our main competitor (x2). Also the categorical variable is store and I have modeled both using this as a variable and without.
-- Here are my model results:
The regression equation is
Gallons = 4.19 - 293 MUSAPriceMinusLowKey + 0.983 MUSA4wkgal Predictor Coef SE Coef T P Constant 4.1932 0.7771 5.40 0.000 MUSAPriceMinusLowKey -292.75 13.07 -22.41 0.000 MUSA4wkgal 0.982941 0.002580 380.92 0.000 S = 47.1163 R-Sq = 91.0% R-Sq(adj) = 91.0% Analysis of Variance Source DF SS MS F P Regression 2 323496065 161748033 72861.37 0.000 Residual Error 14352 31860609 2220 Total 14354 355356674 Source DF Seq SS MUSAPriceMinusLowKey 1 1380837 MUSA4wkgal 1 322115228 Source DF Seq SS PriceMinusLowKey 1 177122 4wkgal 1 8611828 Lack of fit test Possible curvature in variable MUSAPric (P-Value = 0.000 ) Possible interaction in variable MUSA4wkg (P-Value = 0.000 ) Overall lack of fit test is significant at P = 0.000
So I am at my wit's end now and am not sure what to do next. Is it possible that the S shape residual plot is common for a dataset with so many observations and it is not a concern? Thanks in advance for any help you can give.