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Stata Textbook Examples
Applied Linear Statistical Models by Neter, Kutner, et. al.
Chapter 25: Analysis of CovarianceInputting the Cracker Promotion data, p. 1020.clear input y x treat store 38 21 1 1 39 26 1 2 36 22 1 3 45 28 1 4 33 19 1 5 43 34 2 1 38 26 2 2 38 29 2 3 27 18 2 4 34 25 2 5 24 23 3 1 32 29 3 2 31 30 3 3 21 16 3 4 28 29 3 5 endTable 25.1, p. 1020.
table treat store, contents(mean y mean x)---------------------------------------- | store treat | 1 2 3 4 5 ----------+----------------------------- 1 | 38 39 36 45 33 | 21 26 22 28 19 | 2 | 43 38 38 27 34 | 34 26 29 18 25 | 3 | 24 32 31 21 28 | 23 29 30 16 29 ----------------------------------------Fig. 25.5, p. 1021.
twoway (scatter y x if treat==1, ms(oh) msize(large) legend(label(1 "Treatment 1"))) /// (scatter y x if treat==2, legend(label(2 "Treatment 2"))) /// (scatter y x if treat==3, ms(sh) legend(label(3 "Treatment 3")))
Creating the indicator and interaction variables for the Cracker data set. First we need to calculate the overall mean which will be used to generate the x variable (x = X-mean), table 25.2, p. 1021.
quietly sum x gen littlex = x-r(mean) gen I1 = 0 replace I1 = 1 if treat==1 replace I1 = -1 if treat==3 gen I2 = 0 replace I2 = 1 if treat==2 replace I2 = -1 if treat==3 gen I1x = I1*littlex gen I2x = I2*littlex sort treat store list treat store y x littlex I1 I2 I1x I2x mean x +---------------------------------------------------------+ | treat store y x littlex I1 I2 I1x I2x | |---------------------------------------------------------| 1. | 1 1 38 21 -4 1 0 -4 0 | 2. | 1 2 39 26 1 1 0 1 0 | 3. | 1 3 36 22 -3 1 0 -3 0 | 4. | 1 4 45 28 3 1 0 3 0 | 5. | 1 5 33 19 -6 1 0 -6 0 | |---------------------------------------------------------| 6. | 2 1 43 34 9 0 1 0 9 | 7. | 2 2 38 26 1 0 1 0 1 | 8. | 2 3 38 29 4 0 1 0 4 | 9. | 2 4 27 18 -7 0 1 0 -7 | 10. | 2 5 34 25 0 0 1 0 0 | |---------------------------------------------------------| 11. | 3 1 24 23 -2 -1 -1 2 2 | 12. | 3 2 32 29 4 -1 -1 -4 -4 | 13. | 3 3 31 30 5 -1 -1 -5 -5 | 14. | 3 4 21 16 -9 -1 -1 9 9 | 15. | 3 5 28 29 4 -1 -1 -4 -4 | +---------------------------------------------------------+Regressing Y on littlex, I1 and I2, table 25.3, p. 1022. Testing for treatment effect, p. 1023-1024.
regress y littlex I1 I2 predict r, residSource | SS df MS Number of obs = 15 -------------+------------------------------ F( 3, 11) = 57.78 Model | 607.828691 3 202.609564 Prob > F = 0.0000 Residual | 38.5713085 11 3.50648259 R-squared = 0.9403 -------------+------------------------------ Adj R-squared = 0.9241 Total | 646.4 14 46.1714286 Root MSE = 1.8726 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- littlex | .8985594 .1025849 8.76 0.000 .6727716 1.124347 I1 | 6.017407 .7082568 8.50 0.000 4.458544 7.57627 I2 | .9420168 .6986826 1.35 0.205 -.5957732 2.479807 _cons | 33.8 .483493 69.91 0.000 32.73584 34.86416 ------------------------------------------------------------------------------matrix list e(V)symmetric e(V)[4,4] littlex I1 I2 _cons littlex .01052366 I1 .01894258 .50162766 I2 -.01473312 -.26028512 .48815738 _cons 0 0 0 .23376551test I1=I2( 1) I1 - I2 = 0 F( 1, 11) = 17.06 Prob > F = 0.0017Figure 25.6a, p. 1023.
twoway scatter treat r, ylabel(#3)
Figure 25.6b, p. 1023.
qnorm r
Table 25.4, p. 1023.
regress y littlexSource | SS df MS Number of obs = 15 -------------+------------------------------ F( 1, 13) = 5.44 Model | 190.677778 1 190.677778 Prob > F = 0.0364 Residual | 455.722222 13 35.0555556 R-squared = 0.2950 -------------+------------------------------ Adj R-squared = 0.2408 Total | 646.4 14 46.1714286 Root MSE = 5.9208 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- littlex | .7277778 .3120521 2.33 0.036 .0536301 1.401925 _cons | 33.8 1.528737 22.11 0.000 30.49736 37.10264 ------------------------------------------------------------------------------Estimation of treatment effects, pair-wise comparisons, p. 1024.
gen treat1 = 0 replace treat1 = 1 if treat==1 gen treat2 = 0 replace treat2 = 1 if treat==2 gen treat3 = 0 replace treat3 = 1 if treat==3 regress y littlex treat1 treat2 treat3, noconstantSource | SS df MS Number of obs = 15 -------------+------------------------------ F( 4, 11) = 1265.12 Model | 17744.4287 4 4436.10717 Prob > F = 0.0000 Residual | 38.5713085 11 3.50648259 R-squared = 0.9978 -------------+------------------------------ Adj R-squared = 0.9970 Total | 17783 15 1185.53333 Root MSE = 1.8726 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- littlex | .8985594 .1025849 8.76 0.000 .6727716 1.124347 treat1 | 39.81741 .8575507 46.43 0.000 37.92995 41.70486 treat2 | 34.74202 .8496605 40.89 0.000 32.87193 36.61211 treat3 | 26.84058 .8384392 32.01 0.000 24.99518 28.68597 ------------------------------------------------------------------------------ lincom treat1-treat2 ( 1) treat1 - treat2 = 0 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | 5.07539 1.228965 4.13 0.002 2.370456 7.780324 ------------------------------------------------------------------------------ di r(se)^2 1.5103553 lincom treat1-treat3 ( 1) treat1 - treat3 = 0 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | 12.97683 1.205623 10.76 0.000 10.32327 15.63039 ------------------------------------------------------------------------------ di r(se)^2 1.4535275 lincom treat2-treat3 ( 1) treat2 - treat3 = 0 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | 7.901441 1.188746 6.65 0.000 5.285029 10.51785 ------------------------------------------------------------------------------ di r(se)^2 1.4131167Estimating the mean response for each treatment group when X is at its mean (X=25), p. 1026.
Note: This code uses the matrices saved by the regression command.
regress y littlex I1 I2 matrix v = e(V) Source | SS df MS Number of obs = 15 -------------+------------------------------ F( 3, 11) = 57.78 Model | 607.828691 3 202.609564 Prob > F = 0.0000 Residual | 38.5713085 11 3.50648259 R-squared = 0.9403 -------------+------------------------------ Adj R-squared = 0.9241 Total | 646.4 14 46.1714286 Root MSE = 1.8726 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- littlex | .8985594 .1025849 8.76 0.000 .6727716 1.124347 I1 | 6.017407 .7082568 8.50 0.000 4.458544 7.57627 I2 | .9420168 .6986826 1.35 0.205 -.5957732 2.479807 _cons | 33.8 .483493 69.91 0.000 32.73584 34.86416 ------------------------------------------------------------------------------ di _b[_cons]+_b[I1] 39.817407 di v[4,4]+v[2,2]+2*v[4,2] .73539317 di _b[_cons]+_b[I2] 34.742017 di v[4,4]+v[3,3]+2*v[4,3] .72192289 di _b[_cons]-_b[I1]-_b[I2] 26.840576 di v[4,4]+v[2,2]+v[3,3]-2*v[4,2]-2*v[4,3]+2*v[3,2] .7029803Table 25.5 and testing for parallel slopes, in other words, testing to see if the interactions are significant, p. 1027.
regress y littlex I1 I2 I1x I2x Source | SS df MS Number of obs = 15 -------------+------------------------------ F( 5, 9) = 35.11 Model | 614.879165 5 122.975833 Prob > F = 0.0000 Residual | 31.5208354 9 3.50231504 R-squared = 0.9512 -------------+------------------------------ Adj R-squared = 0.9241 Total | 646.4 14 46.1714286 Root MSE = 1.8714 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- littlex | .9387352 .1126656 8.33 0.000 .6838679 1.193603 I1 | 6.269902 .7516696 8.34 0.000 4.569507 7.970297 I2 | .7179135 .7160036 1.00 0.342 -.9017991 2.337626 I1x | .1525057 .1843832 0.83 0.430 -.2645981 .5696094 I2x | .0525185 .145611 0.36 0.727 -.2768765 .3819134 _cons | 33.89433 .5123434 66.16 0.000 32.73533 35.05333 ------------------------------------------------------------------------------ test I1x=I2x=0 ( 1) I1x - I2x = 0 ( 2) I1x = 0 F( 2, 9) = 1.01 Prob > F = 0.4032Inputting the Salable Flowers data set.
clear input y x a b rep 98 15 1 1 1 60 4 1 1 2 77 7 1 1 3 80 9 1 1 4 95 14 1 1 5 64 5 1 1 6 55 4 2 1 1 60 5 2 1 2 75 8 2 1 3 65 7 2 1 4 87 13 2 1 5 78 11 2 1 6 71 10 1 2 1 80 12 1 2 2 86 14 1 2 3 82 13 1 2 4 46 2 1 2 5 55 3 1 2 6 76 11 2 2 1 68 10 2 2 2 43 2 2 2 3 47 3 2 2 4 62 7 2 2 5 70 9 2 2 6 end label var y "yield" label var x "plot size" label var a "variety" label var b "moisture"Table 25.6, p. 1029.
table rep b , contents(mean y mean x) by(a) ---------------------- variety | moisture and rep | 1 2 ----------+----------- 1 | 1 | 98 71 | 15 10 | 2 | 60 80 | 4 12 | 3 | 77 86 | 7 14 | 4 | 80 82 | 9 13 | 5 | 95 46 | 14 2 | 6 | 64 55 | 5 3 ----------+----------- 2 | 1 | 55 76 | 4 11 | 2 | 60 68 | 5 10 | 3 | 75 43 | 8 2 | 4 | 65 47 | 7 3 | 5 | 87 62 | 13 7 | 6 | 78 70 | 11 9 ----------------------Fig. 25.7, p. 1030.
twoway (scatter y x if a==1 & b==1, legend(label(1 "A1B1"))) /// (scatter y x if a==1 & b==2, legend(label(2 "A1B2"))) /// (scatter y x if a==2 & b==1, legend(label(3 "A2B1"))) /// (scatter y x if a==2 & b==2, legend(label(4 "A2B2")))
Generating the variable for X centered at its mean, the indicator variables and their interactions, p. 1028.sum x gen littlex = x - r(mean) gen I1 = 1 replace I1 = -1 if a==2 gen I2 = 1 replace I2 = -1 if b==2 gen I12 = I1*I2Table 25.7, regression output and sums of squares, p. 1030 and the test of the interaction, p. 1031.regress y littlex I1 I2 I12 anova y littlex I1 I2 I12, continuous(littlex)Source | SS df MS Number of obs = 24 -------------+------------------------------ F( 4, 19) = 197.45 Model | 4966.51882 4 1241.6297 Prob > F = 0.0000 Residual | 119.481183 19 6.28848331 R-squared = 0.9765 -------------+------------------------------ Adj R-squared = 0.9716 Total | 5086 23 221.130435 Root MSE = 2.5077 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- littlex | 3.276882 .1300174 25.20 0.000 3.004752 3.549011 I1 | 2.042339 .5210844 3.92 0.001 .9516966 3.132981 I2 | 3.68078 .51291 7.18 0.000 2.607247 4.754313 I12 | .8192204 .51291 1.60 0.127 -.2543125 1.892753 _cons | 70 .511879 136.75 0.000 68.92862 71.07138 ------------------------------------------------------------------------------ Number of obs = 24 R-squared = 0.9765 Root MSE = 2.50768 Adj R-squared = 0.9716 Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 4966.51882 4 1241.6297 197.45 0.0000 | littlex | 3994.51882 1 3994.51882 635.21 0.0000 I1 | 96.6018263 1 96.6018263 15.36 0.0009 I2 | 323.849473 1 323.849473 51.50 0.0000 I12 | 16.0422442 1 16.0422442 2.55 0.1267 | Residual | 119.481183 19 6.28848331 -----------+---------------------------------------------------- Total | 5086 23 221.130435Figure 25.8 p.1031, estimated treatment means plot (x=0).regress y littlex I1 I2 I12 twoway (function y = _b[_cons]+0*_b[littlex]+x*_b[I1]+(-1)*_b[I2]+x*(-1)*_b[I12], range(1 -1) n(2)) /// (function y = _b[_cons]+0*_b[littlex]+x*_b[I1]+(1)*_b[I2]+x*1*_b[I12], range(1 -1) n(2)) /// , xscale(reverse) xlabel(1 "a1" -1 "a2") ylabel(40 100) legend(label(1 "b2") label(2 "b1"))
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