Topic Overview Working-Hotelling Confidence Band Inference Example using SAS ANOVA Table
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Working-Hotelling Confidence Band (1) This gives a confidence limit for the whole line at once, in contrast to the confidence interval for just one Yˆh at a time. Regression line b0 b1 X h describes E Yh for given X h . We have 95% CI for specific X h .
E(Yh ) = Yˆh
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Working-Hotelling Confidence Band (2) We want a 95% Confidence band for all Xh – this is a confidence limit for the whole line at once, in contrast to the confidence interval for ˆ Y just one h at a time.
, The confidence limit is given by 2 W 2F 1 ;2, n 2 . Since we are doing where all values of X h at once, it will be wider at each X h than CIs for individual X h . Yˆh W s Yˆh
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Working-Hotelling Confidence Band (3) We are used to constructing CI’s with t’s, not W’s. Can we fake it? c We can find a new, smaller alpha for t that would give the same results – kind of an “effective alpha” that takes into account that you are estimating the entire line. 2 We find W for our desired true α, and then c t find the effective α to use with t that gives c t W(α) = t (α ). 4-5
SAS Example (musclemass.sas) (Problem 1.27 in KNNL) Muscle mass is expected to decrease with age. Study explores this relationship in women (n = 60) 15 women randomly selected from each of four age groups 40-49, 50-59, 60-69, 70-79 We will analyze this data set assuming that the simple linear regression model applies. 4-6
Read in the Data For textbook files – easiest way is to simply open data as text file or through website and paste it into SAS using “datalines”. DATA muscle; input mmass age; datalines; 106 43 106 41 ..... ; 4-7
Produce a Scatter Plot goptions ftitle=centb ftext=swissb htitle=3 htext=1.5 ctitle=blue ctext=black; symbol1 v=dot c=blue ; axis1 label=('Age (Years)'); axis2 label=(angle=90 'Muscle Mass'); PROC GPLOT data=muscle; plot mmass*age /haxis=axis1 vaxis=axis2; title 'Muscle Mass vs Age in women'; RUN; QUIT;
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Examining Scatter Plots Form – linear looks mostly reasonable Direction – muscle mass seems to decrease as age increases Strength – there is quite a bit of scatter so the relationship is likely weak to moderate
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Regression Model Goals Estimate the difference in mean muscle mass for women differing in age by 1 year. Produce CI’s and PI’s for women age 50, 60, and 70 Plot 95% Confidence Band for the regression line.
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Preliminaries DATA slime; age = 50; mmass = .; output; age = 60; mmass = .; output; age = 70; mmass = .; output; DATA muscle; set muscle slime; PROC PRINT; RUN;
This adds to the data set so that we can easily predict for ages of 50, 60, and 70.
