MEDD8815 Chap.7 Inference for Simple and Multiple Regression
Inference for Simple and Multiple Regression
The line as an estimate
Session 7 returns to the least-squares line of Session 2 and treats it as an estimate of a population line. The simple linear regression model says that for each value of the explanatory variable the responses are Normal, centred on the population line, with the same standard deviation throughout.
The fitted intercept and slope estimate the population values, and the regression standard error, built from the residuals with n minus 2 degrees of freedom, estimates the common standard deviation.
Inference about the slope
The slope has a standard error, and the t statistic that divides the slope by its standard error tests the hypothesis of no linear relationship, with n minus 2 degrees of freedom.
The chapter reproduces the coefficients table and the ANOVA table that software prints, shows how the total variation splits into a regression part and a residual part, and verifies that the ANOVA F statistic equals the square of the slope t statistic, as the session handout notes.
The conditions are checked with residual plots.
Two intervals at one value of x
At a chosen value of the explanatory variable, a confidence interval estimates the mean response of all cases with that value, while a prediction interval covers a single new case and is always wider.
A computed band chart shows both, and the chapter explains the SPSS Save options that write the predicted values, standard errors and interval limits into the data file.
Multiple regression
With several explanatory variables, each slope measures the effect of its variable with the others held constant, and the residual degrees of freedom become n minus p minus 1. The overall F test asks whether any slope is non-zero; individual t tests ask whether each variable is useful given the others.
The chapter uses an AskSia simulated sample in which one predictor is clearly useful alone yet adds little once a correlated predictor is in the model, which is the distinction the session homework is built around.
What this chapter covers
- 01
The simple linear regression model and its conditions
- 02
The regression standard error
- 03
Confidence interval and t test for the slope
- 04
The ANOVA table for regression
- 05
F equals t squared in simple regression
- 06
Interval for a mean response and prediction interval
- 07
The multiple regression model and the overall F test
- 08
Useful alone against useful given the other predictors
Worked example · free
Read a coefficients table and write the conclusion
- 2Compute t and its degrees of freedom, and bracket the P-value.
- 2Build the interval with the matching critical value.
- 2Interpret the slope interval and R Square in context.
Key terms
- Regression Standard Error
- The estimate of the common standard deviation of responses about the population line, the square root of the residual sum of squares divided by n minus 2.
- Slope Standard Error
- The estimated standard deviation of the fitted slope, the regression standard error divided by the square root of the sum of squared deviations of x.
- Mean Response Interval
- A confidence interval for the average response of all cases with a given value of the explanatory variable.
- Prediction Interval
- An interval for the response of a single new case at a given value of the explanatory variable, always wider than the mean response interval.
- Coefficient of Determination
- The proportion of the total variation in the response explained by the regression, the model sum of squares divided by the total sum of squares.
- Multiple Regression
- A regression with two or more explanatory variables, in which each slope gives the change in mean response with the other variables held constant.
- Overall F Test
- The test of the hypothesis that every slope in a regression model is zero, against the alternative that at least one is not.
- Residual Plot
- A plot of residuals against the explanatory variable or fitted values, used to check linearity, constant spread and outliers.
Inference for Simple and Multiple Regression FAQ
Why is the prediction interval wider than the confidence interval for the mean?
The mean response interval only has to account for uncertainty in where the population line lies. A single new case also varies around that line by the full residual standard deviation, which never shrinks however much data you collect, so its interval must be wider.
A predictor is significant on its own but not in the multiple regression. Is that a mistake?
No. A t test in a multiple regression asks whether the variable adds information given the other predictors. If two predictors are correlated, the second may carry little that the first does not, so it can be useful alone and redundant in the joint model.
What does a significant overall F test tell me?
That at least one slope in the model is not zero. It does not tell you which predictors matter or that all of them do; the individual t tests, read with the caution that each is conditional on the others, answer that.
Assessment move
Take one SPSS regression output and annotate every number on it in your own words: what it estimates, what its standard error is for, which test it belongs to and with how many degrees of freedom. Then run the same predictors one at a time and compare. That single exercise covers the session's homework themes more efficiently than working new problems.
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