Basic Practice of Statistics - 3rd Edition
Chapter 21 1
BPS - 5th Ed. Chapter 23 1
Chapter 23
Inference for Regression
BPS - 5th Ed. Chapter 23 2
Objective: To quantify the linear relationship between an explanatory variable (x) and response variable (y).
We can then predict the average response for all subjects with a given value of the explanatory variable.
Linear Regression (from Chapter 5)
BPS - 5th Ed. Chapter 23 3
Case Study
Researchers explored the crying of infants four to ten days old and their IQ test scores at age three to determine if
more crying was a sign of higher IQ
Crying and IQ
Karelitz, S. et al., “Relation of crying activity in early infancy to speech and intellectual development at age three years,”
Child Development, 35 (1964), pp. 769-777.
BPS - 5th Ed. Chapter 23 4
Case Study Crying and IQ Data collection
Data collected on 38 infants Snap of rubber band on foot caused infants
to cry – recorded the number of peaks in the most active 20
seconds of crying (explanatory variable x)
Measured IQ score at age three years using the Stanford-Binet IQ test (response variable y)
BPS - 5th Ed. Chapter 23 5
Case Study Crying and IQ
Data TABLE 23.1
BPS - 5th Ed. Chapter 23 6
Case Study Crying and IQ Data analysis
Scatterplot of y vs. x shows a moderate positive linear relationship, with no extreme outliers or potential influential observations
Basic Practice of Statistics - 3rd Edition
Chapter 21 2
BPS - 5th Ed. Chapter 23 7
Case Study Crying and IQ Data analysis
Correlation between crying and IQ is r = 0.455 (from formula in Chapter 4)
Least-squares regression line for predicting IQ from crying is
(as in Ch. 5) R2 = 0.207, so 21% of the variation in IQ scores
is explained by crying intensity
BPS - 5th Ed. Chapter 23 8
We now want to extend our analysis to include inferences on various components involved in the regression analysis – slope – intercept – correlation – predictions
Inference
BPS - 5th Ed. Chapter 23 9
Conditions required for inference about regression (have n observations on an explanatory variable x and a response variable y) 1. for any fixed value of x, the response y varies
according to a Normal distribution. Repeated responses y are independent of each other.
2. the mean response µy has a straight-line relationship with x: µy = α + βx . The slope β and intercept α are unknown parameters.
3. the standard deviation of y (call it σ) is the same for all values of x. The value of σ is unknown.
Regression Model, Assumptions
BPS - 5th Ed. Chapter 23 10
the regression model has three parameters: α, β, and σ
the true regression line µy = α + βx says that the mean response µy moves along a straight line as x changes (we cannot observe the true regression line; instead we observe y for various values of x)
observed values of y vary about their means µy according to a Normal distribution (if we take many y observations at a fixed value of x, the Normal pattern will appear for these y values)
Regression Model, Assumptions
BPS - 5th Ed. Chapter 23 11
the standard deviation σ is the same for all values of x, meaning the Normal distributions for y have the same spread at each value of x
Regression Model, Assumptions
BPS - 5th Ed. Chapter 23 12
When using the least-squares regression line , the slope b is an unbiased estimator of the true slope β, and the intercept a is an unbiased estimator of the true intercept α
Estimating Parameters: Slope and Intercept
Basic Practice of Statistics - 3rd Edition
Chapter 21 3
BPS - 5th Ed. Chapter 23 13
the standard deviation σ describes the variability of the response y about the true regression line
a residual is the difference between an observed value of y and the value predicted by the least-squares regression line:
the standard deviation σ is estimated with a sample standard deviation of the residuals (this is a standard error since it is estimated from data)
Estimating Parameters: Standard Deviation
BPS - 5th Ed. Chapter 23 14
The regression standard error is the square root of the sum of squared residuals divided by their degrees of freedom (n-2):
Estimating Parameters: Standard Deviation
BPS - 5th Ed. Chapter 23 15
Case Study Crying and IQ
Since , b = 1.493 is an unbiased estimator of the true slope β, and a = 91.27 is an unbiased estimator of the true intercept α – because the slope b = 1.493, we estimate that
on the average IQ is about 1.5 points higher for each added crying peak.
The regression standard error is s = 17.50 – see pages 600-601 in the text for this calculation
BPS - 5th Ed. Chapter 23 16
The most common hypothesis to test regarding the slope is that it is zero: H0: β = 0 – says regression line is horizontal (the mean of
y does not change with x) – no true linear relationship between x and y – the straight-line dependence on x is of no value
for predicting y Standardize b to get a t test statistic:
Hypothesis Tests for Slope (test for no linear relationship)
BPS - 5th Ed. Chapter 23 17
Hypothesis Tests for Slope
the standard error of b is a multiple of the regression standard error:
Test statistic for H0: β = 0 :
– follows t distribution with df = n-2
BPS - 5th Ed. Chapter 23 18
Hypothesis Tests for Slope
P-value: [for T ~ t(n-2) distribution]
Ha: β > 0 : P-value = P(T ≥ t)
Ha: β < 0 : P-value = P(T ≤ t)
Ha: β ≠ 0 : P-value = 2×P(T ≥ |t|)
Basic Practice of Statistics - 3rd Edition
Chapter 21 4
BPS - 5th Ed. Chapter 23 19
Case Study Crying and IQ
Hypothesis Test for slope β
P-value
t = b / SEb = 1.4929 / 0.4870 = 3.07
Significant linear relationship
BPS - 5th Ed. Chapter 23 20
The correlation between x and y is closely related to the slope (for both the population and the observed data) – in particular, the correlation is 0 exactly when
the slope is 0 Therefore, testing H0: β = 0 is equivalent to
testing that there is no correlation between x and y in the population from which the data were drawn
Test for Correlation
BPS - 5th Ed. Chapter 23 21
The t-score simplifies to:
Degrees of freedom: n-2
Test for Correlation
BPS - 5th Ed. Chapter 23 22
There does exist a test for correlation that does not require a regression analysis – Table E on page 695 of the text gives critical
values and upper tail probabilities for the sample correlation r under the null hypothesis that the correlation is 0 in the population look up n and r in the table (if r is negative, look up
its positive value), and read off the associated probability from the top margin of the table to obtain the P-value just as is done for the t table (Table C)
Test for Correlation
BPS - 5th Ed. Chapter 23 23
Case Study Crying and IQ
Test for H0: correlation = 0 Correlation between crying and IQ is r = 0.455 Sample size is n=38 From Table E: for Ha: correlation > 0 , the
P-value is between .001 and .0025 (using n=40) – P-value for two-sided test is between .002 and .005
(matches two-sided P-value for test on slope) – one-sided P-value would be between .005 and .01 if
we were very conservative and used n=30
BPS - 5th Ed. Chapter 23 24
A level C confidence interval for the true slope β is b ± t* SEb – t* is the critical value for the t distribution with
df = n-2 degrees of freedom that has area (1-C)/2 to the right of it
– recall, the standard error of b is a multiple of the regression standard error:
Confidence Interval for Slope
Basic Practice of Statistics - 3rd Edition
Chapter 21 5
BPS - 5th Ed. Chapter 23 25
Case Study Crying and IQ
b=1.4929, SEb = 0.4870, df = n-2 = 38-2 = 36 (df = 36 is not in Table C, so use next smaller df = 30) For a 95% C.I., (1-C)/2 = .025, and t* = 2.042
So a 95% C.I. for the true slope β is: b ± t* SEb = 1.4929 ± 2.042(0.4870) = 1.4929 ± 0.9944 = 0.4985 to 2.4873
Confidence interval for slope β
BPS - 5th Ed. Chapter 23 26
Once a regression line is fit to the data, it is useful to obtain a prediction of the response for a particular value of the explanatory variable ( x* ); this is done by substituting the value of x* into the equation of the line ( ) for x in order to calculate the predicted value
We now present confidence intervals that describe how accurate this prediction is
Inference about Prediction
BPS - 5th Ed. Chapter 23 27
There are two types of predictions – predicting the mean response of all subjects
with a certain value x* of the explanatory variable
– predicting the individual response for one subject with a certain value x* of the explanatory variable
Predicted values ( ) are the same for each case, but the margin of error is different
Inference about Prediction
BPS - 5th Ed. Chapter 23 28
To estimate the mean response µy, use an ordinary confidence interval for the parameter µy = α + βx* – µy is the mean of responses y when x = x* – 95% confidence interval: in repeated samples
of n observations, 95% of the confidence intervals calculated (at x*) from these samples will contain the true value of µy at x*
Inference about Prediction
BPS - 5th Ed. Chapter 23 29
To estimate an individual response y, use a prediction interval – estimates a single random response y rather
than a parameter like µy – 95% prediction interval: take an observation
on y for each of the n values of x in the original data, then take one more observation y at x = x*; the prediction interval from the n observations will cover the one more y in 95% of all repetitions
Inference about Prediction
BPS - 5th Ed. Chapter 23 30
Inference about Prediction
Basic Practice of Statistics - 3rd Edition
Chapter 21 6
BPS - 5th Ed. Chapter 23 31
Both confidence interval and prediction interval have the same form.
– both t* values have df = n-2 – the standard errors (SE) differ for the two
intervals (formulas on previous slide) the prediction interval is wider than the
confidence interval
Inference about Prediction
BPS - 5th Ed. Chapter 23 32
Residual Plots x = number of beers y = blood alcohol
Roughly linear relationship; spread is even across entire data range (‘random’ scatter about zero)
Residuals: -2 731 -1 871 -0 91 0 5578 1 1 2 39 3 (4|1 = .041) 4 1
(close to Normal)
BPS - 5th Ed. Chapter 23 33
Residual Plots ‘x’ = collection of explanatory variables, y = salary of player
Standard deviation is not constant everywhere (more variation among players with higher salaries)
BPS - 5th Ed. Chapter 23 34
Residual Plots x = number of years, y = logarithm of salary of player
A clear curved pattern – relationship is not linear
BPS - 5th Ed. Chapter 23 35
Independent observations – no repeated observations on the same
individual
True relationship is linear – look at scatterplot to check overall pattern – plot of residuals against x magnifies any
unusual pattern (should see ‘random’ scatter about zero)
Checking Assumptions
BPS - 5th Ed. Chapter 23 36
Constant standard deviation σ of the response at all x values – scatterplot: spread of data points about the
regression line should be similar over the entire range of the data
– easier to see with a plot of residuals against x, with a horizontal line drawn at zero (should see ‘random’ scatter about zero) (or plot residuals against for linear regr.)
Checking Assumptions
Basic Practice of Statistics - 3rd Edition
Chapter 21 7
BPS - 5th Ed. Chapter 23 37
Response y varies Normally about the true regression line – residuals estimate the deviations of the
response from the true regression line, so they should follow a Normal distribution make histogram or stemplot of the residuals
and check for clear skewness or other departures from Normality
– numerous methods for carefully checking Normality exists (talk to a statistician!)
Checking Assumptions
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