P Value Insignificant but Residual Plot Great

[jsconte18]

Hi

I have a model that has three parameters: intercept, x1, and x2. 

x1's p value is 0.7730

x2's p value is less than 0.0001

But x1's residual plot looks really good and spread out

while x2's shows a quadratic U shape

I am wondering why this is and what all goes into it. Thank you!

SAS code

DATA groupD17;
INPUT Y X1 X2;
DATALINES;
. . . data goes here . . .
;
PROC REG DATA=groupD17;
MODEL Y = X1 X2;
RUN;

[StatDave]

You probably need to add a quadratic term for X2 in your model. Create a variable: x22=x2**2; and then include it in your MODEL statement.

[KevinScott]

The relationship between y and x1 x2 in your data can be captured by using a regression between y and just the quadratic term.  Assuming X2 is X1*X1, try running.

PROC REG DATA=groupD17;
MODEL Y = X2;
RUN;

And compare the Rsquare with the original model.

Here is an example that will help you understand why a model with only a quadratic term and no linear term can fit well:

data examples(drop=i);
do i=1 to 1000;
x1=10*ranuni(213)-5;
x2=x1*x1;
/*y_only_quad is a difference of squares with no linear term
y_only_quad=x1^2-4 + N(0,1) error*/
y_only_quad=(x1+2)*(x1-2) + rannor(33);
/*y_lin_quad is a quadratic polynomial with a linear term
y_lin_quad=x1^2+4x+4 + N(0,1) error*/
y_lin_quad=(x1+2)*(x1+2) + rannor(89);
output;
end;
run;

ods graphics;
proc sgplot data=examples;
scatter x=x1 y=y_only_quad;
scatter x=x1 y=y_lin_quad;
refline 0 / axis=y;
refline 0 / axis=x;
run;

/*linear term will not be significant assuming alpha level 0.05
rsquare is 0.98*/
proc reg data=examples;
model y_only_quad= x1 x2;
run;
quit;

/*remove linear term rsquare is still 0.98*/
proc reg data=examples;
model y_only_quad= x2;
run;
quit;

/*run model where the linear term is required
the linear term is significant at alpha level 0.05*/
proc reg data=examples;
model y_lin_quad= x1 x2;
run;
quit;