Statistical Inferences

Regression Statistical Inference

Statistical Inferences

What types of inferences can we make?

• Estimate Y given value(s) of X

• Confidence limits on estimates of Y

• Prediction limits on estimates of Y

• Confidence limits on Slope parameter

Estimates of Y

How do we find estimates of Y given X?

Expected value of Y:

E{Y} = E{B0 + B1(X) + e}

since B0, B1, and X are constants…

E{Y} = B0 + B1(X) + E{e}

E{e}=0

So E{Y} = B0 + B1(X)

Just plug the value of X into the regression equation!

Confidence Limits of Y

How do we find confidence limits for estimates of Y?

It is similar to finding confidence limits of a sample mean

2-sided conf limits = E{Y} +/- tinv(%conf, Df) (s){conf}

1-sided lower conf = E{Y} – tinv(%conf, Df) (s){conf}

1-sided upper conf = E{Y} + tinv(%conf, Df) (s){conf}

where :

Df = degrees of freedom for the error term in the model = (n-p)

s{conf} = SQRT( MSE (1/n + (X-Xbar)^2/SSX))

X = value at which Y is to be estimated

Xbar = average of X values

SSX = sum(Xi-Xbar)^2

Confidence Limits of Y

In JASP, you can let the code do the work.

In Excel, you will have to calculate the s{conf} value in order to calculate the

confidence limits of Y.

Prediction Limits of Y

How do we find prediction limits for estimates of Y?

2-sided prediction limits = E{Y} +/- tinv(%conf, Df)(s){pred}

1-sided lower pred limit = E{Y} – tinv(%conf, Df)(s){pred}

1-sided upper pred limit = E{Y} + tinv(%conf, Df)(s){pred}

where :

Df = degrees of freedom for the error term in the model = (n-p)

s{pred} = SQRT( MSE(1+1/n + (X-Xbar)^2/SSX))

X = value at which Y is to be estimated

Xbar = average of X values

SSX = sum(Xi-Xbar)^2

Prediction Limits of Y

In Minitab, again, you can let the code do the work –

Under the “Options” button, just enter the confidence level, the X value,

and check the “Prediction Limit” box to get two-sided confidence limits.

In Excel, you will have to calculate the s{pred} value in order to calculate the

confidence limits of Y.

Confidence Limits on Coefficients

How do we find confidence limits for regression coefficients?

Again, it is similar to finding confidence limits of a sample mean

2-sided conf limits = Coeff +/- tinv(%conf, Df)(std error)

1-sided lower conf = Coeff – tinv(%conf, Df)(std error)

1-sided upper conf = Coeff + tinv(%conf, Df)(std error)

where :

Df = degrees of freedom for the error term in the model = (n-p)

Coeff and Std error comes from the regression results table

Confidence Limits for Coefficients

ANOVA

df SS MS F Significance F

Regression 1 2999.584891 2999.585 3.549601 0.10854555

Residual 6 5070.290109 845.0484

Total 7 8069.875

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 260.278 19.35648165 13.44655 1.05E-05 212.9143234 307.6416

Mileage -0.0021 0.001116634 -1.88404 0.108546 -0.00483609 0.0006285

Excel will calculate the 2-sided confidence limits for you:

• Click the “Confidence Level” box and enter the 2-sided

confidence level value (will have to modify % if problem

is asking for one-sided confidence)