Friedman's nonparametric test
We already know that when we need to compare two or more samples on mean values, we can use nonparametric methods along with parametric ones. These methods are free from a number of limitations of parametric methods, do not require the normal distribution of data and the equality of variance. Their advantage is especially valuable in the case when the dependent variable can not be expressed in a metric scale.
For experimental plans with repeated measurements, there is also a nonparametric method that allows to circumvent the assumptions of the parametric model of variance analysis discussed above. This method is named after its developer  the American economist M. Friedman.
Like any other nonparametric method, Friedman's test assumes that the data analyzed represent the results of the ranking. We already know that in this case the application of nonparametric analysis proves to be optimal. If the experimental results obtained are not rank values, they should be transformed into ranks, which obviously leads to the loss of some information. Therefore, if the dependent variable is represented in a metric scale, the Friedman test turns out to be less powerful than the procedures of variance analysis with repeated measurements, and can only be recommended if the variationalcovariance matrix exhibits strong heterogeneity.
Having the results of ranking for each subject, we calculate the value of the total square of the experimental impact in the standard way, i.e. by the formula (3.2). The total square for the experimental error is estimated as the degree of variability of the data within the subjects. Recall that it includes the variance caused by the difference in the experimental conditions, and the residual dispersion. Since, as a result of using the ranks, the variance between the mean and total values of the dependent variable of all subjects turns out to be zero, since the sum of the ranks for all subjects will be the same, this statistic is equal to the total total square (formula (3.4)). The number of degrees of freedom for this statistic, as indicated in Table. 4.2, is n ( k  1). As a result, we get the following Friedman statistics Q :
(4.8)
If the number of subjects n and the number of levels of the investigated independent variable k turn out to be quite large ( < strong> p & gt; 15 and k & gt; 4), statistics Q is described approximately by the distribution of χ2 with k  1 degrees of freedom (see Appendix 2). Then special tables should be used.
The formula (4.8) is universal for calculating the statistics Q. It can be applied regardless of whether or not there are duplicate ranks in the experiment results. However, in the case where the experimental results do not contain fractional ranks, the total square for the experimental error can be expressed somewhat differently, since its value is determined solely on the basis of the number of experimental conditions and the number of subjects:
Then the formula (4.8) takes the following form:
For pairwise comparison of various experimental conditions, procedures developed, for example, by Shaykh and Hamerli can be used. However, in standard statistical analysis packages, they are usually not available.
Practical examples
Changing the exponents of y in Stevens law depending on the frequency of the sound signals
And. Y. Myshkin [13] carried out a number of interesting experimental studies aimed at studying factors that cause a change in the exponent in the formula of the basic psychophysical law. In particular, he sought the dependence of the exponent γ on the frequency of the sound signal. Directly the value of γ was calculated as the ratio of the logarithm of the value of the sensation, expressed in fractions with respect to the maximum value of the estimate, to the intensity of the sound signal (expressed in decibels). The results of the measurements are shown in Table. 4.5.
Considering these data, the author of the study notes: "With increasing frequency of sound stimuli, the exponent decreases. The relationship between the frequency of the audio signal and the exponent is a nonlinear one. " Unfortunately, the author [13] does not provide any statistics in support of his conclusions. Let's try to fill this gap and answer the question of how statistically reliable the revealed dependence is.
Table 4.5
The individual values of the exponent in in Stevens's law with a different frequency of audio signals (I.Yu. Myshkin [13])
Subject 
Frequency, Hz 

125 
500 
1000 
2000 
4000 
8000 

1 
0.37 
0.33 
0.33 
0.38 
0.30 
0.28 
2 
0.33 
0.33 
0.34 
0.33 
0.29 
0.30 
3 
0.35 
0.35 
0.33 
0.36 
0.28 
0.28 
4 
0.36 
0.34 
0.33 
0.35 
0.30 
0.30 
5 
0.35 
0.31 
0.33 
0.32 
0.29 
0.28 
6 
0.32 
0.31 
0.33 
0.32 
0.29 
0.27 
7 
0.36 
0.35 
0.34 
0.35 
0.28 
0.29 
8 
0.36 
0.33 
0.33 
0.34 
0.29 
0.31 
9 
0.37 
0.33 
0.34 
0.34 
0.30 
0.30 
10 
0.38 
0.35 
0.31 
0.34 
0.31 
0.29 
Average 
0.35 
0.33 
0.33 
0.34 
0.29 
0.29 
We will perform manual calculations using the desktop calculator, or partially manually, using a spreadsheet, for example MS Excel. Calculations will be carried out in accordance with the algorithm presented in Table. 4.3. In this connection it is necessary to find four elements of the formulas:
The results of calculations are as follows: (1) = 6,305; (2) = 6.352; (3) = 6.341; (4) = 6.309. Based on these calculations, we can calculate the total squares and estimate the number of degrees of freedom for each dispersion source (Table 4.6). Now, in order to estimate the statistical effect of the signal frequency, it is sufficient to divide the mean square value for the signal frequency (0.00713) by the residual dispersion value (0.00016). It turns out that F (5, 45) = 44.43. Referring to the statistical tables (see Appendix 4), you can be sure that the result obtained is extremely highly reliable. Thus, the author's conclusion that there are differences in the value of the degree of γ depending on the frequency of the presented sound signals seems to be statistically justified, and its publication would undoubtedly benefit if the author cited the results of our statistical analysis.
Table 4.6
Calculation of the sum of squares, numbers of degrees of freedom and mean squares for different dispersion sources when evaluating the effect of signal frequency
Source variances 
Total square ( SS ) 
Degree of freedom ( df ) 
Average square ( MS ) 
Between subjects 
0.004 
9 
0.00042 
Inside the subjects 
0.043 
50 
0.00086 
Signal frequency 
0.036 
5 
0.00713 
Balance 
0.007 
45 
0.00016 
Common 
0.047 
59 
0.00079 
Estimating the overall signal frequency effect and obtaining statistical support for the opinion of the author of the article [13], we return to his statement that this effect turns out to be nonlinear. To study this issue we divide the total square of the frequency of the signal into two additive components. One of them will determine the linear contrast, the second  nonlinear.
First, we set the following factors that determine linear contrast: 5; 3; 1; 1; 3; 5. Based on these coefficients, you can also determine the value of the linear contrast C lin :
Now we can estimate the average square of this contrast:
Dividing the obtained value by the variance of the statistical error (residual variance), we have F (1, 45) = 150.09; p & lt; 0.001, which indicates an exceptionally highvalue effect. Note that the total square of the signal frequency is 0.036, and its component describing the linear dependence is 0.024. Thus, it turns out that the linear trend describes 2/3 of the variance associated with the effect of the independent variable, while the nonlinear one is only a third: 0.036  0.024 = 0.012. Nevertheless, the nonlinear contrast also turns out to be statistically significant  F (4, 45) = 18.75; p & lt; 0.01.
It should be noted that the assumption of quadratic dependence also provides a reliable statistical solution: F (l, 45) = 7.44; p & lt; 0.05. However, if we carefully consider the nature of the observed dependence, it can be noted that the possible nonlinearity of the effect under investigation is that, as the signal frequency increases, the exponent γ decreases first, then increases, and then decreases again. In other words, we can assume that w is a kind of dependency. Such a relationship must be described by a fourthorder degree.
It turns out that the fourthorder polynomial contrast is indeed quite pronounced and its dispersion reaches a value of 0.005. This value is statistically reliable: F (1, 45) = 32.14; p & lt; 0.01. Thus, the conclusion about the nonlinearity of the revealed effect also has statistical grounds.
It should be noted that when calculating the statistics F , we used as the variance of the statistical error the residual dispersion, the same as when estimating the overall effect. Nevertheless, another strategy for making a statistical decision is not without foundation. In this case, as the magnitude of the error, statistics are considered in which the difference between the total square of the basic experimental effect and the sum of all its components from the first (linear) to the considered order is added to the residual dispersion value. For example, to calculate the linear trend error SS dev lin, the following relationship should be defined:
Substituting this expression available to us the value we get SS dev lin = 0,007 + (0,036  0.024) = 0.019. This statistic has n ( k 1) 1 degrees of freedom. In our case, this number turns out to be 49. Consequently, the mean square corresponding to this statistic is 0.019/49 = 0.000388. Thus, we can adjust the value of F for the linear component  F (1, 59) = 61.93 ; p & lt; 0.01.
Similarly, adjust the error variance for the fourthorder component SS dev quant :
where quad , cubic, quart mean the second, third, and fourth order polynomials.
>Get
This statistic has n ( k  1)  4 degrees of freedom. Then the mean square for this error will be 0.0128/46 = 0.000278. Statistics F slightly decreases, but still remains statistically highly reliable  F (1, 46) = 17, 97; p & lt; 0.01.
Finally, we recall that the best way to estimate the variance of an experimental error is to calculate it using formula (4.7). This is the way to calculate, as a rule, modern statistical packages.
As we see, two different ways of component estimation give us grounds to make two completely different conclusions. And although the conclusion about the linear dependence is somewhat more statistically justified, a fourthorder nonlinear dependence is also quite possible.
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