The chi-square statistic ((n-p-1)/n)Q is commonly used in practice and referred to an m-DF

chi-square distribution to compute multitrait association test's p values, which can lead to significantly inflated type I errors at stringent genomewide significance levels.

Totals Market Results from Domed and Retractable Roof Stadia: 2011-2014 Setting Overs Under s Pushes Over Win Percentage Dome 85 74 1 52.46% Retractable Roof 81 75 3 51.92% Dome and 166 149 4 52.70% Retractable Roof Other Games 482 467 11 50.79% Setting Log Log Likelihood Likelihood - Fair Bet - No Profits (50%) (52.4%) Dome 0.7616 0.0743 Retractable Roof 0.2308 NA Dome and 0.9179 0.0128 Retractable Roof Other Games 0.2371 NA The log likelihood test statistics have a

chi-square distribution with one degree of freedom.

For some distributions, like t distribution or

Chi-square distribution, the t or Chi-square value depends on a single value of degree of freedom only.

When [n.sub.t] is large enough, it is possible to use the Central Limit Theorem to approximate the

Chi-square distribution to a Gaussian distribution [31], and the following approximation holds

where is [x.sup.2.sub.1-[alpha],df] the [alpha]th percentile of the

chi-square distribution with df, degrees of freedom, n is the sample size, df = n - m (number of independent random samples) is degrees of freedom defined as the number of values that are free to vary, and Z(1-P)/2 is the pth percentile of the standard normal distribution.

If we denote [u.sup.1] = b'Sb/[[tau].sup.2], [u.sup.2] = e'e/[[tau].sup.2], then [u.sub.1] ~ [[chi square].sub.f] ([[lambda].sub.1]), and [u.sub.2] ~ [[chi square].sup.n-k] for given [tau], where [[lambda].sub.1] = [beta]'S[beta]/[[tau].sup.2], [[chi square].sub.k] (lambda) is the non central

chi-square distribution with f degrees of freedom and noncentrality parameter [lambda].

Moreover, Cochran's theorem has shown that V [??] (N - 1) v/[[sigma].sup.2] has a

chi-square distribution with df - N - 1 degrees of freedom [18].

The distinct advantage of the prescribed methods is that it circumvents the uncertainty of sample variance by taking account of the underlying

chi-square distribution of sample variance and permits a corrected sample size determination according to the desired assurance probability and expected power considerations.

The calculated statistics is compared with the critical value of the

chi-square distribution. The critical value is read out from the statistical tables for the specific degree of freedom ((k-1)*(l-1)) and also the significance level (probability of rejecting a true hypothesis).

Results indicate a noncentral

chi-square distribution for rows and columns of the GxE interaction matrix, which was also verified by the Kolmogorov-Smirnov test and Q-Q plot.

An assumption is also made when using the

chi-square distribution as an approximation to the distribution of kh2, is that the frequencies expected under independence should not be "too small".

The square of a normal N(0, 1) variable has the

chi-square distribution [[chi].sub.(1).sup.2] with degrees of freedom 1.