For convenience, some of these tests may be performed interactively on sigular data sets.

**t-test****U-test****Kolomogorov-Smirnov****Fisher exact test****Grubbs test****2x2 Cross tables****Neumann trend test****Correlation coefficient r => p-value****Probability => Correlation coefficient r**

"... the Kolmogorov-Smirnov test (K-S test or KS test) is a nonparametric test of the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution (one-sample K-S test), or to compare two samples (two-sample K-S test). ...

... The two-sample K-S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples. ..."

Use this utilitiy to test equality of your data.

In

The KS-test dialog opens up:

Fill in data values for group1 and group2.

Or copy Tab/NewLine/space separated values into ClipBoard.

Click

(

Click the

Results are shown in the

The Result table indicates:

Top: | Results from KS-test |

Middle: | For comparison - results from t-test |

Bottom: | Some descriptive statistics |

Additionally are shown:

- density function of pasted data
- Empirical Distribution Function (EDF) of pasted data (~cumulative density function)

To experiment with KS-test, try

2 identical gaussians | Two different randomly samples Gaussian distributed data sets. Both with Mean=0, SDev=1 |

2 gaussians different mean | Two different randomly samples Gaussian distributed data sets. Both with different Mean, similar SDev |

2 gaussians, different SDev | Two different randomly samples Gaussian distributed data sets. Both with similar Mean, different SDev |

2 different gaussians | Two different randomly samples Gaussian distributed data sets. Both with different Mean and SDev |

Gaussian <=> Random | A Gaussian distributed data sets versus a random dataset. |

Random <=> Random | Two different random dataset. |

For more details about KS-test see Non parametric tests | Kolmogorov-Smirnov test.

A simple example:

Analyse the handedness of members ina group of individual depending ongender:

Handedness | ||||
---|---|---|---|---|

Left-handed | right-handed | Sum | ||

Gender | Female | 44 | 4 | 48 |

Male | 43 | 9 | 52 | |

Sume | 87 | 13 | 100 |

Very often 2x2 tables are used to compare predicted paramters versius observed paramters.

Here thea are called

Predicted | |||
---|---|---|---|

Posiivte | Negative | ||

Observed | Positive | True Positive | False Negative |

Negative | False Positive | True Negative |

With the values TP,FP,FN,TN we can derive various values to indicate association betweeen the two variables (Predicted/Observed):

TP | True positives (Hits) |

FP | False positives (Class 1 error) |

FN | False negatives (Class 2 error) |

TN | True negatives (Correct rejection) |

SPC | Specificity = TN / (FP+TN) = True Negative Rate (TNR) = Selectivity |

SEN | Sensitivity = TP / (TP+FN) = True Positive Rate (TPR) = Recall = Hit rate |

PPV | Positive Predictive Value = TP / (TP+FP) = Precision |

NPV | Negative Predictive Value = TN / (TN+FN) |

PRE | Prevalence = (TP+FN)/(TP+FP+FN+TN) |

ACC | Accuracy = (TP+TN)/(TP+FP+FN+TN) = Random precision |

BA | Balance Accuracy = (TPR+TNR)/2 |

FPR | False Positive Rate = FP / (FP+TN) = fall out |

FNR | False Negative Rate = FN ( FN+TP) = Miss rate |

Gain | PPV / ACC |

MCC | Matthews Correlation Coefficient = ((TP*TN)-(FP*FN)) / sqrt((TP+FP)*(FN+TN)*(TP+FN)*(FP*TN)) = phi-Coefficient |

FDR | False Dicovery Rate = FP / (FP+TP) |

FOR | False Omission Rate = FN / (FN+TN) |

LR+ | Positive Likelihood ratio = TPR/FPR = Senistivity/FPR |

LR- | Negative Likelihood ratio = FNR/TNR = FNR/Specificity |

FS | F-Score = 2*TP / (TP+FP+TP+FN) = F1-Score or F Measure |

RR | Relative Risk = TP/(TP+FP) / FN/(FN+TN) |

DOP | difference between disproportion = | TP/(TP+FP) - FN/(FN+FP) |

PT | Prevalence Threshold = sqrt(FPR) / (sqrt(TPR)+sqrt(FPR)) |

TS | Threat Score = Critical Succes Index (CSI = TP / (TP+FN+FP) |

FM | Fowlkes-Mallows Index = sqrt(PPV*TPR) |

BM | Bookmarker Informdness = Informdness = TPR + TNR -1 |

MK | MArkness = DeltaP = PPS+NPV-1 |

Odds ratio | (TP/FP) / (FN/TN)' |

pDOF | Significance by Pearson''s Goodness-of-Fit Test' |

pFET | Significance by Fisher''s Exact test |

Yule's Q | Yule coefficient of association = (TP-TN) / (TP+TN) |

YuleS's Y | coefficient of colligation = = 1 - sqrt( 1-sqr(1-YulesQ)) / YulesQ |

Additionally, several Similarity measures< - as used in clustering of binary verctors - can be computed:

Simple Matching

Russel-Rao

Tanimoto

Kulczynski

Braun

Hamann<

Cohen's Kappa

Bandigwala

Ochiai

Phi

Sneath

Simpson

Yule

Accuracy

F1-Score

From

A new Winwow opens up:

Fill in the values und press

In a first step we can compute a t-value:

r should be in the range: -1 < r < 1, n > 2

(adopted from: Miles and Banyard's (2007), Understanding and Using Statistics in Psychology --- A Practical Introduction)

From t-distribution we can find the corresponding p-value.

In

Enter your data (

Click

Edit

In a first step, we can get the t-value from an inverse t-distrubtion for the desired p-value.

Applying and converting the above formula we can compute

In

Enter your data(

Click

Edit

The Mantel test, named after Nathan Mantel, is a statistical test of the correlation between two matrices. The matrices must be of the same dimension; in most applications, they are matrices of interrelations between the same vectors of objects.

Originally the Mantel test was introduced to compare distance matrices: square matrices with identical dimensions and positive data values.

But the tst may be also performed with not squared matrices.

The two matrices must have:

- Same dimensions (i.e. same number of rows/columns).

In case the two matrices have different dimensions,truncates the larger one to the dimension of the smaller one.*SUMO* - Positive data values

negative data values are processed too, but the test result may be meaningless - All data cells should contain numbers.

converts non numeric data or empty data cells to ZERO*SUMO* - Numbers hould be supplied in international format (decimal-point as divider)

tries to convert german format (decimal-comma)*SUMO*

Mantel's test statistic is comupted with a basic cross product formula:

With

A normalized correlation value is computed:

with ‾x, ‾y = average from matrix x and y respectively.

s

- r=1 : highest similarity, matrices are identical
- r~0 : just random values, no similarity
- r=-1 : matrices are contradictory

A significance value for similarity is computed based on a permutation scheme wher rows and columns of (one) matrix are radomly shuffeld.

For each of the n

The number m of permutations where r' < r is counted and onverted into a p-value:

p = (m+1) / n

To extract a trustful p-value, the number of permutations should be adopted to the critical p-value:

p | n^{p} |
---|---|

0.05 | 1000 |

0.01 | 5000 |

0.001 | 50000 |

... | ... |

With

In the parameter dialog select/specify:

- Data matrix files: either type the names, separated by semicolon

click the ... button to open a file selection box

drag files from file explorer into the data-matrix field - Header rows/columns: define number of such rows/coloms, containing description/annotions not useful for computation

Header rows/colums MUST be identical in the matrices - Number of permutation cycle for computation of p-value