StatsVariancesTest

StatsVariancesTest [/ALPH=val /Q/Z/METH=m /WSTR=strList ] [wave1, wave2,... wave100 ]

The StatsVariancesTest operation performs Bartlett's or Levene's test to determine if wave variances are equal. Output is to the W_StatsVariancesTest wave in the current data folder or optionally to a table.

Flags

/ALPH=val	Sets the significance level (default val =0.05).
/DEST=vWave	Specify the output wave for the variances test. If you do not specify this flag, the operation saves the output in the wave W_StatsVariancesTest in the current data folder.
	This flag was added in Igor Pro 10.00.
/FREE	Creates the specified destination wave as a free wave.
	/FREE is allowed only in functions, and only if the destination waves are simple names or wave reference structure fields.
	See Free Waves for more discussion.
	The /FREE flag was added in Igor Pro 10.00.
/METH=m	Specifies the test type.
	m =0: Bartlett test (default). m =1: Levene's test using the mean. m =2: Modified Levene's test using the median. m =3: Modified Levene's test using the 10% trimmed mean.
/Q	No results printed in the history area.
/T=k	Displays results in a table. k specifies the table behavior when it is closed.
	k =0: Normal with dialog (default). k =1: Kills with no dialog. k =2: Disables killing.
	The table is associated with the test, not the data. If you repeat the test, it will update any existing table with the new results.
/WSTR=waveListString
	Specifies a string containing a semicolon-separated list of waves that contain sample data. Use waveListString instead of listing each wave after the flags.
/Z	Ignores errors. V_flag will be set to -1 for any error and to zero otherwise.

Details

All tests define the null hypothesis by

$$\displaystyle H_{0}: \quad \sigma_{1}^{2}=\sigma_{2}^{2}=\ldots=\sigma_{k}^{2},$$

agains the alternative

$$\displaystyle H_{a}: \quad \sigma_{i}^{2} \neq \sigma_{j}^{2} \text { for at least one } i \neq j .$$

Bartlett's test computes:

$$T=\frac{(n-k) \ln \left(\sigma_{w}^{2}\right)-\displaystyle \sum_{i=1}^{k}\left(n_{i}-1\right) \ln \left(\sigma_{i}^{2}\right)}{1+\displaystyle \frac{1}{3(k-1)}\left[\sum_{i=1}^{k} \frac{1}{n_{i}-1}-\frac{1}{N-k}\right]} .$$

where

$$\displaystyle \sigma_{i}^{2}$$

is the variance of the i th wave, N is the sum of the points of all the waves, n_i is the number of points in wave i, and k is the number of waves. The weighted variance is given by

$$\displaystyle \sigma_{w}^{2}=\sum_{i=1}^{k} \frac{\left(n_{i}-1\right) \sigma_{i}^{2}}{N-k}$$

H₀ is rejected if T is greater than the critical value taken from the

$$\displaystyle \chi^{2}$$

distribution computed by solving for x:

$$\displaystyle 1- { alpha }=1-{ gammq }\left(\frac{k-1}{2}, \frac{x}{2}\right) .$$

Levene's test computes:

$$W=\frac{(N-k) \displaystyle \sum_{i=1}^{k} n_{i}\left(\bar{Z}_{i}-\bar{Z}\right)^{2}}{(k-1) \displaystyle \sum_{i=1}^{k} \sum_{j=1}^{k}\left(Z_{i j}-\bar{Z}_{i}\right)^{2}},$$

where

$$\begin{array}{l} Z_{i j}=\left|Y_{i j}-\bar{Y}_{i}\right|, \\ \\ \displaystyle \bar{Z}_{i}=\frac{1}{n_{i}} \sum_{j=1}^{k} Z_{i j}, \\ \\ \displaystyle \bar{Z}=\frac{1}{N} \sum_{i=1}^{k} \sum_{j=1}^{k} Z_{i j} . \end{array}$$ $$\displaystyle \bar{Y}_{i}$$

depends on /METH.

H₀ is rejected if W is greater than the critical value from the F distribution computed by solving for x:

$$\displaystyle 1-\text { alpha }=1-\text { betai }\left(\frac{v_{2}}{2}, \frac{v_{1}}{2}, \frac{v_{2}}{v_{2}+v_{1} x}\right) .$$

References

NIST/SEMATECH, Bartlett's Test, in NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm, 2005.

See Also

Statistical Analysis

Demos

Open Variances-Test Demo.pxp