Comparison of noncentral and central distributions (original) (raw)

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Online calculator for critical values and cumulative probabilities

z and noncentral distributions (χ2,t, and F)

Noncentral χ2

Let Z i,i_=0,1,2,... denote a series of independent random variables of standard normal distribution. V = \sum_{i=1}^{df}{Z_i}^2 will be a random variable of χ2 distribution with df degrees of freedom. For any given series of constants μ_i,_i_=1,2,...,df, \sum_{i=1}^{df}(Z_i+\mu_i)^2 will be a random variable of the respective noncentral χ2 distribution with the same df and the distinct noncetral parameter ncp = \sum_{i=1}^{df}{\mu_i}^2. It is different from the random variable V + ncp = \sum_{i=1}^{df}{Z_i}^2 + \sum_{i=1}^{df}{\mu_i}^2 of the respective central χ2 distribution with a central drift.

Noncentral t

For any given constant μ0, \frac{Z_0+\mu_0}{\sqrt{V/df }} =\frac{Z_0+\mu_0}{\sqrt{\sum_{i=1}^{df}{Z_i}^2/df}} is a random variable of noncentral t-distribution with noncentrality parameter μ0, which is different from \frac{Z_0}{\sqrt{V/df\ }}+\mu_0, the central t-distributed random variable drifted with the same mean.

If df on this display is set to Inf and noncentral parameter set to 0, a standard normal distribution will be produced and critical z score calculated.

Noncentral F

The noncentral parameter of F is only defined on its numerator. The noncentral F distributed \frac{\sum_{i=1}^{df_1}(Z_i+\mu_i)^2/df_1}{\sum_{i=df_1+1}^{df_1+df_2}Z_i^2/df_2} with noncentral parameter \sum_{i=1}^{df_1}\mu_i^2 is different from the central F distributed random variable plus the respective constant, \frac{\sum_{i=1}^{df_1}Z_i^2/df_1}{\sum_{i=df_1+1}^{df_1+df_2}Z_i^2/df_2}+\frac{\sum_{i=1}^{df_1}\mu_i^2}{df_1} .

Power of t test for a given Cohen's δ

Example of one-group mean test

A normally distributed population, for example, IQ distribution of students, is sampled N (=) times independently. The mean and standard deviation estimates from all M samples are respectively denoted M and S D in the current replication.

The statistical interest is usually on the mean of population, named μ; sometimes also on the standard deviation of population, named σ. The statistic t is defined as following --

t:=\frac{M-\mu_{null}}{SD/\sqrt{N}}

It measures whether or not M is significantly a baseline μ_n_ u l l_=, relative to the scale of standard error estimate of M. If μ is really μ_n u l l, the t statistic distribution is known with noncentral parameter 0 and degrees freedom (N − 1). Type I error, denoted α(=), defines the probability domain of the extreme t values.

However, the real μ may be μ_a_ l t e r n a t i v e rather than μ_n_ u l l. Then, the noncentral parameter of the t statistic distribution will change to be

N^{1/2}\times\frac{\mu_{alternative}-\mu_{null}}{\sigma}

wherein δ: = (μ_a_ l t e r n a t i v e − μ_n_ u l l) / σ is estimated by Cohen's d: = (M − μ_n_ u l l) / S D. A known/hypothesized δ, eg. , together with the sample size N, will give a known/hypothesized noncentral t distribution, while a μ_a_ l t e r n a t i v e alone without a given σ is helpless.

Change M (=) and S D(=), then verify whether they affect the statistical power.

Power of F test for a given Cohen's _f_2

Power of SEM close-fit test for a given RMSEA