options pagesize=60 linesize=80; data scenic; infile 'scenic.dat' firstobs=2; input Stay Age Risk Culture Chest Beds School Region Census Nurses Facil; Nratio = Nurses / Census ; R1 = -(Region-4)*(Region-3)*(Region-2)/6; R2 = (Region-4)*(Region-3)*(Region-1)/2; R3 = -(Region-4)*(Region-2)*(Region-1)/2; S1 = School-1; proc reg data=scenic; model Risk = S1 Culture Stay Nurses Nratio { R1 R2 R3 } Chest Beds Census Facil / selection=stepwise groupnames = 'School' 'Culture' 'Stay' 'Nurses' 'Nratio' 'Region' 'Chest' 'Beds' 'Census' 'Facil'; run ;
EDITED SAS OUTPUT (Complete output)
Stepwise Procedure for Dependent Variable RISK Step 1 Group Culture Entered R-square=0.312659 C(p)=58.3641 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 3.19789965 0.19376813 339.64905575 272.37 0.0001 --- Group Culture -- 62.96314170 50.49 0.0001 CULTURE 0.07325862 0.01030975 62.96314170 50.49 0.0001 ----------------------------------------------------------- Step 2 Group Stay Entered R-square=0.45040256 C(p)=26.82419 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.80549102 0.48775579 2.74400250 2.73 0.1015 --- Group Culture --- 33.39687778 33.19 0.0001 CULTURE 0.05645147 0.00979843 33.39687778 33.19 0.0001 --- Group Stay --- 27.73884588 27.57 0.0001 STAY 0.27547211 0.05246473 27.73884588 27.57 0.0001 ----------------------------------------------------------- Step 3 Group Facil Entered R-square = 0.4934 C(p)=18.3545 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.49133226 0.48163614 0.97401801 1.04 0.3099 --- Group Culture --- 30.59827862 32.69 0.0001 CULTURE 0.05419997 0.00947933 30.59827862 32.69 0.0001 --- Group Stay --- 16.47664606 17.60 0.0001 STAY 0.22390748 0.05336561 16.47664606 17.60 0.0001 --- Group Facil --- 8.65883687 9.25 0.0029 FACIL 0.01963027 0.00645392 8.65883687 9.25 0.0029 --------------------------------------------------------- Step 4 Group Nratio Entered R-square=0.52548 C(p)=12.5433 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -0.49505513 0.59376426 0.61507231 0.70 0.4063 --- Group Culture --- 22.84513509 25.82 0.0001 CULTURE 0.04818092 0.00948204 22.84513509 25.82 0.0001 --- Group Stay --- 21.44995791 24.24 0.0001 STAY 0.26758404 0.05434637 21.44995791 24.24 0.0001 --- Group Nratio --- 6.46014750 7.30 0.0080 NRATIO 0.79262357 0.29333869 6.46014750 7.30 0.0080 --- Group Facil --- 6.75349077 7.63 0.0067 FACIL 0.01747585 0.00632554 6.75349077 7.63 0.0067 --------------------------------------------------------- Step 5 Group Chest Entered R-square=0.53792 C(p)=11.513 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -0.76804342 0.61022741 1.37763165 1.58 0.2109 --- Group Culture --- 16.71979631 19.23 0.0001 CULTURE 0.04318856 0.00984976 16.71979631 19.23 0.0001 --- Group Stay --- 14.43814950 16.60 0.0001 STAY 0.23392650 0.05741114 14.43814950 16.60 0.0001 --- Group Nratio --- 4.38883521 5.05 0.0267 NRATIO 0.67240318 0.29931440 4.38883521 5.05 0.0267 --- Group Chest --- 2.50619510 2.88 0.0925 CHEST 0.00917860 0.00540681 2.50619510 2.88 0.0925 --- Group Facil --- 7.45710068 8.57 0.0042 FACIL 0.01843860 0.00629673 7.45710068 8.57 0.0042 --------------------------------------------------------- Step 6 Group Region Entered R-square=0.56826 C(p)=10.1269 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -0.66156855 0.68931767 0.77004723 0.92 0.3394 --- Group Culture --- 19.41848300 23.23 0.0001 CULTURE 0.04717749 0.00978882 19.41848300 23.23 0.0001 --- Group Stay --- 18.64724032 22.31 0.0001 STAY 0.28408192 0.06015054 18.64724032 22.31 0.0001 --- Group Nratio --- 1.86769604 2.23 0.1380 NRATIO 0.47735146 0.31936579 1.86769604 2.23 0.1380 --- Group Region --- 6.10861501 2.44 0.0689 R1 -0.91152625 0.33831556 6.06877293 7.26 0.0082 R2 -0.61170886 0.30630883 3.33408744 3.99 0.0484 R3 -0.54005754 0.30531855 2.61565335 3.13 0.0799 --- Group Chest --- 3.10587423 3.72 0.0566 CHEST 0.01029102 0.00533912 3.10587423 3.72 0.0566 --- Group Facil --- 7.66252029 9.17 0.0031 FACIL 0.01883340 0.00622080 7.66252029 9.17 0.0031 -------------------------------------------------------- Step 7 Group School Entered R-square=0.5783 C(p)= 9.68028 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -1.29313397 0.79443852 2.18445103 2.65 0.1066 --- Group School --- 2.02343484 2.45 0.1203 S1 0.45874175 0.29282732 2.02343484 2.45 0.1203 --- Group Culture --- 21.14238169 25.64 0.0001 CULTURE 0.05016596 0.00990650 21.14238169 25.64 0.0001 --- Group Stay --- 19.90843811 24.15 0.0001 STAY 0.29583936 0.06020399 19.90843811 24.15 0.0001 --- Group Nratio --- 1.42881407 1.73 0.1909 NRATIO 0.42026288 0.31924279 1.42881407 1.73 0.1909 --- Group Region --- 7.09035688 2.87 0.0402 R1 -0.99737538 0.34041455 7.07745167 8.58 0.0042 R2 -0.64425716 0.30489819 3.68115979 4.46 0.0370 R3 -0.59950685 0.30557155 3.17349874 3.85 0.0525 --- Group Chest --- 2.85453005 3.46 0.0656 CHEST 0.00987802 0.00530873 2.85453005 3.46 0.0656 --- Group Facil --- 9.68526975 11.75 0.0009 FACIL 0.02391008 0.00697611 9.68526975 11.75 0.0009 ---------------------------------------------------------- Step 8 Group Nratio Removed R-square=0.57121 C(p)=9.40791 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -0.83240584 0.71570292 1.12313185 1.35 0.2475 --- Group School --- 2.46231681 2.97 0.0880 S1 0.50274483 0.29193670 2.46231681 2.97 0.0880 --- Group Culture --- 23.66688888 28.50 0.0001 CULTURE 0.05233635 0.00980270 23.66688888 28.50 0.0001 --- Group Stay --- 18.47964968 22.26 0.0001 STAY 0.27469386 0.05822575 18.47964968 22.26 0.0001 --- Group Region --- 9.68716458 3.89 0.0111 R1 -1.10696516 0.33123989 9.27275385 11.17 0.0012 R2 -0.76673818 0.29137725 5.74922078 6.92 0.0098 R3 -0.75936643 0.28139304 6.04647398 7.28 0.0081 --- Group Chest --- 3.92124933 4.72 0.0320 CHEST 0.01132621 0.00521177 3.92124933 4.72 0.0320 --- Group Facil --- 11.30278424 13.61 0.0004 FACIL 0.02545939 0.00690031 11.30278424 13.61 0.0004 ---------------------------------------------------------- All groups of variables left in the model are significant at the 0.1500 level. No other group of variables met the 0.1500 significance level for entry into the model. Summary of Stepwise Procedure for Dependent Variable RISK Group Number Partial Model Step Entered Removed In R**2 R**2 C(p) F Prob>F 1 Culture 1 0.3127 0.3127 58.3641 50.4918 0.0000 2 Stay 2 0.1377 0.4504 26.8242 27.5690 0.0000 3 Facil 3 0.0430 0.4934 18.3545 9.2513 0.0029 4 Nratio 4 0.0321 0.5255 12.5433 7.3012 0.0080 5 Chest 5 0.0124 0.5379 11.5130 2.8818 0.0925 6 Region 8 0.0303 0.5683 10.1269 2.4357 0.0689 7 School 9 0.0100 0.5783 9.6803 2.4542 0.1203 8 Nratio 8 0.0071 0.5712 9.4079 1.7330 0.1909
COMMENTS ON OUTPUT
Up to now our theory has been used to compute P-values or fix critical points to get desired levels. We have assumed that all our null hypotheses are True. I now discuss power or Type II error rates of our tests. Read Chapter 26, section 4, 5 and 6.
Consider a t-test of .
The test statistic is
Table B.5 on page 1346 gives the probability that the absolute value of a non-central t exceeds a given level. If we take the level to be the critical point for a t test at some level then the probability we look up is the corresponding power, that is, the probability of rejection. Notice that the power depends on two unknown quantities, and and on 1 quantity which is sometimes under the experimenter's control (in a designed experiment) and sometimes not (as in an observational study.)
Same idea applies to any linear statistic of the form
- you get a non-central t distribution on
the alternative. So, for example, if testing
but in fact
the non-centrality parameter is
Before an experiment is run it is sensible, if the experiment is costly, to try to work out whether or not it is worth doing. You will nly do an experiment if the probability of Type I and II errors are both reasonably low. The simplest case arises when you prespecify a level, say and an acceptable probability of Type II error, say 0.10. Then you need to specify
The simplest example of the power of an F test arises in regression through the
origin (that is, a model with no intercept term.) Consider the model
The numerator, however, is
FACT:
If W is a
random vector and Q is idempotent with rank pthen WTQW has a non-central
distribution with non-centrality parameter
FACT
If U and V are independent
variables with degrees of freedom
and ,
V is central and U is non-central with non-centrality
parameter
then
POWER CALCULATIONS
Table B 11 gives powers of F tests for various small numerator degrees of freedom and a range of denominator degrees of freedom for or . In the table is simply our (that is, the square root of what I called the non-centrality parameter divided by the square root of 1 more than the numerator degrees of freedom.)
SAMPLE SIZE CALCULATIONS
Sometimes done with charts and sometimes with tables; see table B 12.
This table depends on a quantity