Comparison of noncentral and central distributions

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Online calculator for critical values, cumulative probabilities, and critical noncentral parameters

Input

Choose a statistic: t, χ2, or F (noncentral parameter of χ2 or F must be non-negative)
noncentral parameter (''ncp'')

= ( vs. )

degree freedom

=

degree freedom of denominator (only for F)

=

Mark the points with cumulative probability

=

and the critical statistic

=


Click to update display with precision

Results of critical statistic, cumulative probability, and critical noncentral parameter

How to cite this result

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

Noncentral χ2

Let Zi,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 μi,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

ncp = 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 \infty (Inf in R) and noncentral parameter set to 0, a standard normal distribution will be plotted 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

ncp=\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} .

Confidence interval of standardized effect size by noncentral parameters

Confidence interval of unstandardized effect size like difference of means 1 − μ2) can be found in common statistics textbooks and software, while confidence intervals of standardized effect size, especially Cohen's \tilde{d}:=\frac{\mu_1-\mu_2}{\sigma} and \tilde{f}^2:=\frac{SS(\mu_1,\mu_2,...,\mu_K)}{K \cdot \sigma^2}, rely on the calculation of confidence intervals of noncentral parameters (ncp).

A common method to find confident interval limits of ncp is to solve the critical ncp value for marginal extreme quantile. The ncp parameter of the black curve in the above diagram could be directly adopted. For example, \left(-\infty,8.968\right) can be 97.5% one-way confidence interval of ncp if observed tdf = 4 = 5.1, while change quantile from .025 to .975, we shall find that the two-way interval (1.139, 8.968) can be of 95% confidence level.

T test for mean difference of single group or two related groups

In case of single group, M (μ) denotes the sample (population) mean of single group , and SD (σ) denotes the sample (population) standard deviation. N is the sample size of the group. T test is used for the hypothesis on the difference between mean and a baseline μbaseline. Usually, μbaseline is zero, while not necessary. In case of two related groups, the single group is constructed by difference in each pair of samples, while SD (σ) denotes the sample (population) standard deviation of differences rather than within original two groups.


t:=\frac{M}{SD/\sqrt{N}}=\frac{\sqrt{N}\frac{M-\mu}{\sigma} + \sqrt{N}\frac{\mu-\mu_{baseline}}{\sigma}}{\frac{SD}{\sigma}}
ncp=\sqrt{N}\frac{\mu-\mu_{baseline}}{\sigma} and Cohen's d:=\frac{M-\mu_{baseline}}{SD} is the point estimate of \frac{\mu-\mu_{baseline}}{\sigma}.

So,

\tilde{d}=\frac{ncp}{\sqrt{N}}.

T test for mean difference between two independent groups

n1 or n2 is sample size within the respective group.

t:=\frac{M_1-M_2}{SD_{within}/\sqrt{\frac{n_1 n_2}{n_1+n_2}}}, wherein SD_{within}:=\sqrt{\frac{SS_{within}}{df_{within}}}=\sqrt{\frac{(n_1-1)SD_1^2+(n_2-1)SD_2^2}{n_1+n_2-2}}.
ncp=\sqrt{\frac{n_1 n_2}{n_1+n_2}}\frac{\mu_1-\mu_2}{\sigma} and Cohen's d:=\frac{M_1-M_2}{SD_{within}} is the point estimate of \frac{\mu_1-\mu_2}{\sigma}.

So,

\tilde{d}=\frac{ncp}{\sqrt{\frac{n_1 n_2}{n_1+n_2}}}.


One-way ANOVA test for mean difference across multiple independent groups

One-way ANOVA test applies noncentral F distribution. While with a given population standard deviation σ, the same test question applies noncentral chi-square distribution.

F:=\frac{\frac{SS_{between}}{\sigma^{2}}/df_{between}}{\frac{SS_{within}}{\sigma^{2}}/df_{within}}

For each j-th sample within i-th group Xi,j, denote M_{i}\left(X_{i,j}\right):=\frac{\sum_{w=1}^{n_{i}}X_{i,w}}{n_{i}};\;\mu_{i}\left(X_{i,j}\right):=\mu_{i};.

While,

SS_{between} \over \sigma^{2}

= \frac{SS\left(M_{i}\left(X_{i,j}\right);i=1,2,\cdots,K,\; j=1,2,\cdots,n_{i}\right)}{\sigma^{2}}
= SS\left(\frac{M_{i}\left(X_{i,j}-\mu_{i}\right)}{\sigma}+\frac{\mu_{i}}{\sigma};i=1,2,\cdots,K,\; j=1,2,\cdots,n_{i}\right)
\sim \chi^{2}\left(df=K-1,\; ncp=SS\left(\frac{\mu_{i}\left(X_{i,j}\right)}{\sigma};i=1,2,\cdots,K,\; j=1,2,\cdots,n_{i}\right)\right)

So, both ncp(s) of F and χ2 equate

SS\left(\mu_i(X_{i,j})/\sigma;i=1,2,\cdots,K,\; j=1,2,\cdots,n_{i}\right).

In case of n:=n_1=n_2=\cdots=n_K for K independent groups of same size, the total sample size is N:=n\cdot K.

Cohen's\;\tilde{f}{}^{2}:=\frac{SS(\mu_{1},\mu_{2},...,\mu_{K})}{K\cdot\sigma^{2}}=\frac{SS\left(\mu_i\left(X_{i,j}\right)/\sigma;i=1,2,\cdots,K,\; j=1,2,\cdots,n_{i}\right)}{n\cdot K}=\frac{ncp}{n\cdot K}=\frac{ncp}N.

T-test of pair of independent groups is a special case of one-way ANOVA. Note that noncentral parameter ncpF of F is not comparable to the noncentral parameter ncpt of the corresponding t. Actually, ncp_F=ncp_t^2, and \tilde{f}=\left|\frac{\tilde{d}}{2}\right| in the case.

RMSEA of Structural Equation Model

ncp of χ2 reported by Structural Equation Model softwares is proportional to the population value of RMSEA2, or the squared distance per df from population var-cov matrix to the model space.

\tilde{RMSEA}=\sqrt{\frac{ncp}{(N-1)df}}

Power vs. Standardized Effect Size or ncp

Power of t test for a given Cohen's δ

Example of one-group mean test

Input

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 SD 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 :μnull=,

relative to the scale of standard error estimate of M. If μ is really μnull, 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 μalternative rather than μnull. Then, the noncentral parameter of the t statistic distribution will change to be

\sqrt{N}\times\frac{\mu_{alternative}-\mu_{null}}{\sigma}

wherein δ: = (μalternative − μnull) / σ is estimated by Cohen's d: = (M − μnull) / SD. A known/hypothesized eg.

δ = ,

together with the sample size N, will give a known/hypothesized noncentral t distribution, while a μalternative alone without a given σ is helpless.

Change

M= and SD=,

then verify whether they affect the statistical power.

Results
How to cite this result

Two-related-group mean test

For two-related-group case, the difference scores between each pair of samples can apply one-group mean test interface. Usually μnull is set to zero.

Power of F test for a given Cohen's \tilde{f}^2

Let's use \tilde{f}^2 denote the population of Cohen's f2, specially

SS(\mu_1,\mu_2,\cdots,\mu_K) \over {K\times \sigma^2}

in one-way ANOVA of K groups setup with within-group sample size n and within-group population mean \mu_1,\mu_2,\cdots,\mu_K respectively. The noncentral parameter of the corresponding F or χ2 distribution is ncp=n\times K \times \tilde{f}^2.

Power of SEM close-fit test for a given RMSEA

(N-1)\cdot df\cdot RMSEA^2+df\sim \chi^2_{df,ncp=(N-1)\cdot df\cdot Distance^2perDf(\tilde{\Sigma},Model\,Space)}
RMSEA=\sqrt{\frac{\hat{\chi}^2-df}{df\cdot(N-1)}}

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Comparison of noncentral and central distributions. (yyyy, Month dd). In SlideWiki. Retrieved MM:SS, Month dd, yyyy, from http://mars.wiwi.hu-berlin.de/mediawiki/slides/index.php/Comparison_of_noncentral_and_central_distributions

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