CALM also makes it possible to develop custom codes to test specific hypotheses. The examples below use custom codes in the context of a General Linear Model to test the associated hypotheses.
For the sake of example, this site uses the following data. For reference, the group means are displayed.
mtcars
## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1
## Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4
## Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2
## Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2
## Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4
## Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4
## Merc 450SE 16.4 8 275.8 180 3.07 4.070 17.40 0 0 3 3
## Merc 450SL 17.3 8 275.8 180 3.07 3.730 17.60 0 0 3 3
## Merc 450SLC 15.2 8 275.8 180 3.07 3.780 18.00 0 0 3 3
## Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4
## Lincoln Continental 10.4 8 460.0 215 3.00 5.424 17.82 0 0 3 4
## Chrysler Imperial 14.7 8 440.0 230 3.23 5.345 17.42 0 0 3 4
## Fiat 128 32.4 4 78.7 66 4.08 2.200 19.47 1 1 4 1
## Honda Civic 30.4 4 75.7 52 4.93 1.615 18.52 1 1 4 2
## Toyota Corolla 33.9 4 71.1 65 4.22 1.835 19.90 1 1 4 1
## Toyota Corona 21.5 4 120.1 97 3.70 2.465 20.01 1 0 3 1
## Dodge Challenger 15.5 8 318.0 150 2.76 3.520 16.87 0 0 3 2
## AMC Javelin 15.2 8 304.0 150 3.15 3.435 17.30 0 0 3 2
## Camaro Z28 13.3 8 350.0 245 3.73 3.840 15.41 0 0 3 4
## Pontiac Firebird 19.2 8 400.0 175 3.08 3.845 17.05 0 0 3 2
## Fiat X1-9 27.3 4 79.0 66 4.08 1.935 18.90 1 1 4 1
## Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.70 0 1 5 2
## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2
## Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.50 0 1 5 4
## Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.50 0 1 5 6
## Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.60 0 1 5 8
## Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.60 1 1 4 2
mtcars$cyl <- factor(mtcars$cyl)
summary(lm(mpg~cyl,data=mtcars))
##
## Call:
## lm(formula = mpg ~ cyl, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.2636 -1.8357 0.0286 1.3893 7.2364
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.6636 0.9718 27.437 < 2e-16 ***
## cyl6 -6.9208 1.5583 -4.441 0.000119 ***
## cyl8 -11.5636 1.2986 -8.905 8.57e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.223 on 29 degrees of freedom
## Multiple R-squared: 0.7325, Adjusted R-squared: 0.714
## F-statistic: 39.7 on 2 and 29 DF, p-value: 4.979e-09
tapply(mtcars$mpg,mtcars$cyl,mean)
## 4 6 8
## 26.66364 19.74286 15.10000
Here, the custom hypotheses a specified and then CALM is used to
create the proper codes.
GM <- c(1/3,1/3,1/3)
H1 <- c(1,-1/2,-1/2)
H2 <- c(0,1,-1)
ex_custom.orthogonal <- rbind(GM,H1,H2)
custom.orthogonal <- calm.encode(ex_custom.orthogonal)
custom.orthogonal
## GM H1 H2
## GROUP 1 1 0.6667 0.0
## GROUP 2 1 -0.3333 0.5
## GROUP 3 1 -0.3333 -0.5
contrasts(mtcars$cyl) <- custom.orthogonal[,-1]
summary(lm(mpg~cyl,data=mtcars))$coefficients
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.501856 0.5935308 34.542197 4.104586e-25
## cylH1 9.242208 1.2251189 7.543927 2.574044e-08
## cylH2 4.642857 1.4920048 3.111825 4.152209e-03
In this custom example, the intercept is equivalent to the mean of the group means. The first parameter represents the difference between the first group and mean of the subsequent groups. The second parameter represents the difference between the second and third groups.
Again, the custom hypotheses a specified and then CALM is used to
create the proper codes.
GM <- c(1,0,0)
H1 <- c(1,-1/2,-1/2)
H2 <- c(1,-1,0)
ex_custom.nonorthogonal <- rbind(GM,H1,H2)
custom.nonorthogonal <- calm.encode(ex_custom.nonorthogonal)
custom.nonorthogonal
## GM H1 H2
## GROUP 1 1 0 0
## GROUP 2 1 0 -1
## GROUP 3 1 -2 1
contrasts(mtcars$cyl) <- custom.nonorthogonal[,-1]
summary(lm(mpg~cyl,data=mtcars))$coefficients
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 26.663636 0.9718008 27.437347 2.688358e-22
## cylH1 9.242208 1.2251189 7.543927 2.574044e-08
## cylH2 6.920779 1.5583482 4.441099 1.194696e-04
In this custom example, the intercept is equivalent to the mean of the first group. The first parameter represents the difference between the first group and mean of the subsequent groups. The second parameter represents the difference between the first and second groups.