Test operations analyze the input data to examine the validity of a specific hypothesis.
Common tests involves a computation of some numeric value, also known as "test statistic", which is usually compared with a critical value
in order to determine if you should accept or reject the test hypothesis (H_{0}). Most tests compute a critical value for the given significance alpha
(default value 0.05 or specified via the /ALPH flag). Some tests compute the P-value which you can then directly compare to the desired significance value.

Critical values have been traditionally published in tables for various significance levels and tails of distributions. They are by far the most difficult technical aspect in implementating statistical tests. Critical values are usually obtained from the inverse of the CDF for the appropriate distribution function, i.e., by solving the equation

where alpha is the significance. In some distributions (e.g., Friedman's) the calculation of the CDF is so computationally intensive that it can be impractical to compute for very large parameters. Fortunately, when parameters become large one can usually find an adequate approximate solution. Whenever possible Igor's tests provide exact critical values as well as the common relevant approximations.

Comparison of critical values with published table values can sometimes be interesting as there does not appear to be a standard for determining the critical value when the CDF takes a finite number of discreet values (step-like). In this case the CDF most likely attains the value (1-alpha) in a vertical transition so one could use the x-value for the vertical transition as a critical value or the x-value of the subsequent vertical transition. Some tables reflect a "convervative" approach and print the x-value of subsequent transitions.

Statistical test operations can print some of their results to the history window and save them in a wave in the current data folder. Result waves have a fixed name associated with the operation. Elements in the wave are designated by dimension labels. You can use the /T flag to display the results of the operation in a table with dimension labels. The argument for this flag determines what happens when you kill the table. You can use the /Q in all test operations to restrict printing information in the history window and you can use the /Z flag to make sure that the operations do not report errors except by setting the V_Flag variable to -1.

Statistical test operations tend to include several variations of the named test. You can usually choose to execute one or more variations by specifying the appropriate flags. The following table can be used as a guide for identifying the operation associated with a given test name.

Statistical TestsOperation Name | Common Test Name |

StatsAngularDistanceTest | Performs non-parametric tests on the angular distance between sample data and reference directions for two or more samples contained in individual waves. Angular distance is defined as the shortest distance between two points on a circle (expressed in radians). See also Angular Distance Test Example. |

StatsANOVA1Test | Performs a one-way ANOVA test (fixed-effect model) and optionally the Brown and Forsythe test computing the F'' statistic and its associated degrees of freedom. See also ANOVA1 Test Example. |

StatsANOVA2NRTest | Performs a two-factor analysis of variance (ANOVA) on the data that has no replication, i.e., there is only a single datum for every level of each factor. See also ANOVA2(NR) Test Example. |

StatsANOVA2RMTest | Performs analysis of variance (ANOVA) on the data where replicates consist of multiple measurements on the same subject (repeated measures). See also ANOVA2(RM) Test Example. |

StatsChiTest | The test computes a Chi-squared statistic for comparing two distributions or a Chi-squared statistic for comparing a sample distribution with its expected values. In both cases the comparison is made on a bin-by-bin basis. See also Chi-Squared Test Example. |

StatsCircularCorrelationTest | Performs parametric or non-parametric tests for angular-angular and angular-linear correlations. The non-parametric test (/NAA) follows Fisher and Lee's modification of Mardia's statistic which is an analogue of Spearman's rank correlation. The parametric test for angular-angular correlation (/PAA) involves computation of a correlation coefficient (raa) and then evaluating the mean and variance of equivalent correlation coefficients computed from the same data but each time deleting a different pair of angles. The parametric test for angular-linear correlation (/PAL) involves compuation of the correlation coefficient (ral) which is then compared with a critical value from Chi-squared distribution. See also Angular-Correlation Test Example. |

StatsCircularMeans | Calculates the mean of a number of circular means. The results are the mean angle (grand mean), the length of the mean vector and optionally confidence interval around the mean angle. Other options include performing non-parametric second order analysis (Moore's version of Rayleigh's test), as well as parameteric second order analysis. See also Circular Means Test Example. |

StatsCircularMoments | Computes circular statistical moments and optionally performs angular uniformity tests for the input data. The extent of the calculation is determined by the requested moment. Optional flags include testing the uniformity of the distribution for ungrouped data using Kuiper statistic, the Rayleigh test for uniformity and computing linear order statistics. See also Circular Moments Test Example. |

StatsCircularTwoSampleTest | Performs second order analysis of angles. Using the appropriate flags you can choose between parametric or non-parametric, unordered or paired tests. The non-parametric paired-sample test (/NPR) is Moore's test for paired angles applied in second order analysis. The non-parametric second-order two-sample test (/NSOA) consists of pre-processing where the grand mean is subtracted from the two inputs followed by application of Watson's U2 test. The parametric paired-sample test (/PPR) is due to Hotelling. In this test the input should consist of both angular and vector length data. The test statistic is compared with a critical value from the F-distribution. The parametric second order two-sample test (/PSOA) is an extension of Hotelling one-sample test to second order analysis. See also Circular Two Sample Test Examples. |

StatsCochranTest | Performs Cochran's (Q) test on a randomized block or repeated measures dichotomous data. The operation computes Cochran's statistic and compares it to a critical value from a Chi-squared distribution which interestingly enough depends only of the significance level and the number of groups (columns). The Chi-square distribution is appropriate when there are at least 4 columns and at least 24 total data. See also Cochran Test Example. |

StatsDunnettTest | Performs the Dunnett test of comparing multiple groups to a control group. See also Dunnett Test Example. |

StatsFriedmanTest | Performs Friedman's test on a randomized block of data. The test is a non-parametric analysis of data contained in either individual 1D waves or in a single 2D wave. See also Friedman Test Example. |

StatsFTest | Performs the F-test on the two distributions contained in wave1 and wave2. The waves can be of any real numeric type. They can have arbitrary number of dimensions but they must contain at least two data points each. See also F Test Example. |

StatsHodgesAjneTest | Performs the Hodges-Ajne non-parametric test for uniform distribution around a circle. See also Hodges-Ajne Test Example. |

StatsJBTest | Performs the Jarque-Bera test for normality on numeric data in a single wave. See also Jarque-Bera simulation Example. |

StatsKendallTauTest | Performs the non-parametric Mann-Kendall test which computes a correlation coefficient τ (somewhat similar to Spearman's corelation) from the relative order of the ranks of the data. See also Kendall Tau Examples. |

StatsKSTest | Performs the Kolmogorov-Smirnov (KS) goodness-of-fit test for comparing two continuous distributions. The first distribution is contained in the input wave. The second distribution can be expressed either as the optional wave or as a user function. See also Kolmogorov-Smirnov Examples. |

StatsKWTest | Performs the non-parametric Kruskal-Wallis test which examines variances using the ranks of the data. See also Kruskal-Wallis Examples. |

StatsLinearCorrelationTest | Performs correlation tests on the input waves. Results include the linear correlation coefficient with its standard error, the statistics t and F, Fisher's Z and its critical value. See also Linear Correlation Examples. |

StatsLinearRegression | Performs regression analysis on the input wave(s). Options include: Dunnett's multi-comparison test for the elevations and Tukey-type tests on multiple regressions. See also Linear Regression Examples. |

StatsMultiCorrelationTest | Performs various tests on multiple correlation coefficients. These include multiple comparisons with a control, multiple contrasts test and a Tukey-type multi comparison testing among the correlation coefficients. See also Multi-Correlations Examples. |

StatsNPMCTest | Performs a number of non-parametric multiple comparison tests. These include: Dunn-Holland-Wolfe test, Student-Newman-Keuls test and the Tukey-type (Nemenyi) multiple comparison test. The results are saved in the current data folder in the wave(s) corresponding to the optional flags. You can perform one or more of the supported tests depending on your choice of flags. Note that some tests are only appropriate when you have the same number of samples in all groups. This operation usually follows StatsANOVA1Test or StatsKWTest. See also Non-Parametric Multiple Comparison Examples. |

StatsNPNominalSRTest | Performs a non-parametric serial randomness test for nominal data consisting of two types. The null hypothesis of the test is that the data are randomly distributed. See also Serial Randomness Test. |

StatsRankCorrelationTest | Performs Spearman's rank correlation test. The operation ranks the two inputs and then computes the sum of the squared differences of ranks for all rows. Ties are handled by assigning an average rank and computing the corrected Spearman rank correlation coefficient with ties. See also Spearman Rank Correlation. |

StatsResample | BootStrap and Jacknife tests: Resamples the input wave by drawing (with replacement) values from the input and storing them in the wave W_Resampled. Flag options allow you to iterate the process and to compute various statistics on the drawn samples. See also Bootstrap and Jacknife Examples. |

StatsScheffeTest |
Performs Scheffe's test for the equality of the means. The operation supports two basic modes. The default consists of testing all possible combinations of pairs of waves.
The second mode tests a single combination where the precise form of H_{0} is determined by the coefficients of a contrast wave.
See also Scheffe's Test Example. |

StatsSRTest | Performs a parametric or non-parametric serial randomness tests. The null hypothesis of the tests are that the data are randomly distributed. The parametric test for serial randomness is due to Young and the critical value is obtained from Mean Square Successive Difference distribution. The non-parametric test consists of counting the number of runs which are successive positive or successive negative differences between sequential data. If two sequential data are the same the operation computes two numbers of runs by considering the two possibilities where the equality is replaced with either a positive or a negative difference. The results of the operation include the number of runs up and down, the number of unchanged values the size of the longest run and its associated probability, the number of converted equalities and the probability that the number of runs is less than or equal to the reported number. A separate option in this operation is to run Marsaglia's GCD test on the input. See also Serial Randomness Tests. |

StatsTTest | Performs two kinds of T-tests: The first kind compares the mean of a distribution with a specified mean value and the second T-test compares the means of the two distributions contained in wave1 and wave2. See also T-Test Examples. |

StatsTukeyTest | Performs multiple comparison Tukey (HSD) test and optionally the Newman-Keuls test. See also Tukey-Test Examples. |

StatsVariancesTest | Performs Bartlett's or Levene's test to determine if the variances of the different waves are equal. See also Variances-Tests Examples. |

StatsWatsonUSquaredTest |
Performs Watson's non-parametric two-sample U^{2} test for samples of circular data. The Watson U^{2} H_{0} postulates that the two samples came from the same population against the different populations alternative.
The operation ranks the two inputs while accounting for possible ties. It then computes the test statistic U^{2} and compares it with the critical value.
See also Watson U2 Test.Watson U2 Test.Watson U^{2} Test. |

StatsWatsonWilliamsTest |
Performs the Watson-Williams test for two or more sample means.
The Watson-Williams H_{0} postulates the equality of the means from all samples against the simple inequality alternative.
The test involves the computation of the sums of the sine and cosine of all data from which a weighted r value (rw) is computed. According to Mardia, you
should use different statistics depending on the size of rw: if rw>0.95 it is safe to use the simple F-statistic, while for 0.95>rw>0.7 you should use the F-statistic with the K correction factor.
Otherwise you should use the t-statistic. The operation computes both the (corrected) F-statistic and the t-statistic as well as their corresponding critical values.
See also Watson-Williams Test. |

StatsWheelerWatsonTest | Performs the non-parametric Wheeler-Watson test for two or more samples which postulates that the samples came from the same population. The extension of the test to more than two samples is due to Mardia. The test is not valid for data with ties. See also Wheeler-Watson Test. |

StatsWilcoxonRankTest | Performs the non-parametric Wilcoxon-Mann-Whitney two-sample rank test or the Wilcoxon Signed Rank test on data contained in the waves waveA and waveB. See also Wilcoxon Tests. |

StatsWRCorrelationTest | Performs a Weighted Rank Correlation test. The input waves contain the ranks of sequential factors. The test computes a Top-Down correlation coefficient using Savage sums. See also Weighted Rank Correlation. |

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