Chow's test is for differences between two or more regressions. Assuming that errors in regressions 1 and 2 are normally distributed with zero mean and homoscedastic variance, and they are independent of each other, the test of regressions from sample sizes \(n_1\) and \(n_2\) is then carried out using the following steps. 1. Run a regression on the combined sample with size \(n=n_1+n_2\) and obtain within group sum of squares called \(S_1\). The number of degrees of freedom is \(n_1+n_2-k\), with \(k\) being the number of parameters estimated, including the intercept. 2. Run two regressions on the two individual samples with sizes \(n_1\) and \(n_2\), and obtain their within group sums of square \(S_2+S_3\), with \(n_1+n_2-2k\) degrees of freedom. 3. Conduct an \(F_{(k,n_1+n_2-2k)}\) test defined by $$F = \frac{[S_1-(S_2+S_3)]/k}{[(S_2+S_3)/(n_1+n_2-2k)]}$$ If the \(F\) statistic exceeds the critical \(F\), we reject the null hypothesis that the two regressions are equal.

In the case of haplotype trend regression, haplotype frequencies from combined data are known, so can be directly used.

`chow.test(y1,x1,y2,x2,x=NULL)`

y1

a vector of dependent variable

x1

a matrix of independent variables

y2

a vector of dependent variable

x2

a matrix of independent variables

x

a known matrix of independent variables

The returned value is a vector containing (please use subscript to access them):

the F statistic

the numerator degree(s) of freedom

the denominator degree(s) of freedom

the p value for the F test

Chow GC (1960). Tests of equality between sets of coefficients in two linear regression. Econometrica 28:591-605

# NOT RUN { dat1 <- matrix(c( 1.2, 1.9, 0.9, 1.6, 2.7, 1.3, 3.5, 3.7, 2.0, 4.0, 3.1, 1.8, 5.6, 3.5, 2.2, 5.7, 7.5, 3.5, 6.7, 1.2, 1.9, 7.5, 3.7, 2.7, 8.5, 0.6, 2.1, 9.7, 5.1, 3.6), byrow=TRUE, ncol=3) dat2 <- matrix(c( 1.4, 1.3, 0.5, 1.5, 2.3, 1.3, 3.1, 3.2, 2.5, 4.4, 3.6, 1.1, 5.1, 3.1, 2.8, 5.2, 7.3, 3.3, 6.5, 1.5, 1.3, 7.8, 3.2, 2.2, 8.1, 0.1, 2.8, 9.5, 5.6, 3.9), byrow=TRUE, ncol=3) y1<-dat1[,3] y2<-dat2[,3] x1<-dat1[,1:2] x2<-dat2[,1:2] chow.test.r<-chow.test(y1,x1,y2,x2) # }