Statistic Library / Reference
Omikron Basic on the Internet: http://www.berkhan.de
General -turn page- Table of Contents 
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Statistic Library / Reference |
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Omikron Basic on the Internet: http://www.berkhan.de |
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| This section will serve to explain the procedures and functions of the Statistic Library. However, it is not possible to illustrate the theoretical background of each individual command at this point. If you are working with the Library more often, it will probably be unavoidable to purchase a statistics methodology book. We recommend the book by SACHS (see bibliography), which explains a vast variety of statistical methods in a very practically oriented way. |
| 2.1 Log in and Log off | ||||
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| Stat_Exit | |
| Call this procedure one time at the end of your program. After that you cannot use the Statistic Library anymore. |
| Statistic | |
| A copyright message of the Statistic Library is displayed. |
| 2.2 Calculating Basic Statistics | ||||||||
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| FN Variance#(&X#(),N) | |
| X#(1:N) | Individual values of sample. |
| N | Number of values. |
| Calculates the variance of the sample X#(). | |
| FN St_Dev#(&X#(),N) | |
| X#(1:N) | Individual values of sample. |
| N | Number of values. |
| Calculates die standard deviation. | |
| Mean_Variance &X#(),N,R Mean#,R Var# | |
| X#(1:N) | Individual values of sample. |
| N | Number of values. |
| Mean# | Mean of the sample. |
| Var# | Variance of the sample. |
| Calculates the mean and the variance of the individual values contained in X#(1) to X#(N). | |
| FN Mean_Sample#(&X#(,),N) | |
| X#(1:N,0:1) | Individual values in X#(1:N,0)and respective frequency in X#(1:N,1). |
| N | Number of values. |
| Calculates the weighted mean of the N sample values. | |
| FN Variance_Sample#(&X#(,),N) | |
| X#(1:N,0:1) | Individual values in X#(1:N,0)and respective frequency in X#(1:N,1). |
| N | Number of values. |
| Calculates the weighted variance of the N sample values. | |
| FN St_Dev_Sample#(&X#(,),N) | |
| X#(1:N,0:1) | Individual values in X#(1:N,0)and respective frequency in X#(1:N,1). |
| N | Number of values. |
| Calculates the weighted standard deviation of the N sample values. | |
| FN Sigma_Approx#(Stdev#,N) | |
| Stdev# | Standard deviation. |
| N | Number of values. |
| The true standard deviation of a normal population distribution yields a biased result due to the empirical variance calculated using the above functions. For N > 10, this function will correct the bias. | |
| FN Variation_Coeff#(Stdev#,Mean#) | |
| Stdev# | Standard deviation. |
| Mean# | Mean. |
| Calculates the coefficient of variation, i.e., the standard deviation in units of the arithmetical means. | |
| FN Variation_Coeff_Rel#(Stdev#,Mean#,N) | |
| Stdev# | Standard deviation. |
| Mean# | Mean. |
| N | Number of values. |
| Calculates the relative coefficient of variation, i.e., the coefficient of variation in percent. | |
| FN Mean_Geo#(&X#(),N) | |
| X#(1:N) | Individual values of sample. |
| N | Number of values. |
| Calculates the geometric mean of the N individual values. | |
| FN Mean_Harm#(&X#(),N) | |
| X#(1:N) | Individual values of sample. |
| N | Number of values. |
| Calculates the harmonic mean of the N individual values. | |
| FN Mean_Harm_Sample#(&X#(,),N) | |
| X#(1:N,0:1) | Individual values in X#(1:N,0)and respective frequency in X#(1:N,1). |
| N | Number of values. |
| Calculates the weighted harmonic mean of the N individual values. | |
| 2.3 Distribution and Test Functions |
| Please Note: It is not possible to calculate all distribution and test functions accurately. Especially the inverse functions, which are important for tests, quite often have to be determined through zero algorithms. However, the functions may be used without any problems within the framework of the usual tables and return function values, which are all accurate to more than 3 decimal places, which suffices for normal applications. |
| FN Standard#(X#) | |
| X# | Variable. |
| Calculates the expectation value of the standard normal distribution. | |
| FN Standard_D#(X#) | |
| X# | Variable. |
| Calculates the probability density of the standard normal distribution. | |
| FN Standard_Inv#(P#) | |
| P# | Variable (0<P#<1). |
| Calculates the P# quantile of the standard normal distribution, i.e., its inverse function. Zero is returned if it was not possible to perform the calculation. | |
| FN Normal#(X#,Mu#,Var#) | |
| X# | Variable. |
| Mu# | Mean. |
| Var# | Variance. |
| Calculates the expectation value of the general normal distribution. | |
| FN Normal_D#(X#,Mu#,Var#) | |
| X# | Variable. |
| Mu# | Mean. |
| Var# | Variance. |
| Calculates die probability density of the general normal distribution. | |
| FN Normal_Inv#(P#,Mu#,Var#) | |
| P# | Variable (0<P#<1). |
| Mu# | Mean. |
| Var# | Variance. |
| Calculates the P# quantile of the general normal distribution, i.e., its inverse function. | |
| FN Student#(I,X#) | |
| I | Number of degrees of freedom. |
| X# | Variable. |
| Calculates the expectation value of the Student distribution (t-distribution). | |
| FN Student_D#(I,X#) | |
| I | Number of degrees of freedom. |
| X# | Variable. |
| Calculates the probability density of the Student distribution (t-distribution). | |
| FN Student_Inv#(I,P#) | |
| I | Number of degrees of freedom. |
| P# | Variable (0<P#<1). |
| Calculates the P# quantile of the Student distribution (t-distribution), i.e., its inverse function. Zero is returned, if it was not possible to perform the calculation. | |
| FN Chi2#(I,X#) | |
| I | Number of degrees of freedom(I>=1). |
| X# | Variable (X#>=1). |
| Calculates the expectation value of the chi square distribution. Zero is returned, if it was not possible to perform the calculation. | |
| FN Chi2_D#(I,X#) | |
| I | Number of degrees of freedom. |
| X# | Variable. |
| Calculates the probability density of the chi square distribution. | |
| FN Chi2_Inv#(I,P#) | |
| I | Number of degrees of freedom (I>=1). |
| P# | Variable (0<P#<1). |
| Calculates the P# quantile of the chi square distribution, i.e., its inverse function. Zero is returned, if it was not possible to perform the calculation. | |
| FN Fisher#(I1,I2,X#) | |
| I1 | Number of degrees of freedom in numerator (I1>=1). |
| I2 | Number of degrees of freedom in denominator (I2>=1). |
| X# | Variable. |
| Calculates the expectation value of the Fisher distribution (F-distribution). If it was not possible to perform the calculation, zero is returned. | |
| FN Fisher_D#(I1,I2,X#) | |
| I1 | Number of degrees of freedom in numerator. |
| I2 | Number of degrees of freedom in denominator. |
| X# | Variable. |
| Calculates the probability density of the Fisher distribution (F-distribution). | |
| FN Fisher_Inv#(I1,I2,P#) | |
| I1 | Number of degrees of freedom in numerator (I1>=1). |
| I2 | Number of degrees of freedom in denominator (I2>=1). |
| P# | Variable (0<P#<1). |
| Calculates the P# quantile of the Fisher distribution (F-distribution), meaning its inverse function. If it was not possible to perform the calculation, zero is returned. | |
| FN Expo#(X#,Mu#) | |
| X# | Variable. |
| Mu# | Mean. |
| Calculates the expectation value of the exponential distribution. | |
| FN Expo_Inv#(P#,Mu#) | |
| P# | Variable (0<P#<1). |
| Mu# | Mean. |
| Calculates the P# quantile of the exponential distribution, i.e., its inverse function. | |
| FN Binomial#(X,N,P#) | |
| X | Desired number of elements with character A. |
| N | Total number of elements. |
| P# | Constant probability of success. |
| Calculates the expectation value of the binomial distribution. | |
| FN Binomial_D#(X,N,P#) | |
| X | Desired number of elements with character A. |
| N | Total number of elements. |
| P# | Constant probability of success. |
| Calculates the probability density of the binomial distribution. | |
| FN Hypergeo#(X,N,Sx,Sn) | |
| X | Desired number of elements with character A. |
| N | Number of elements with character A. |
| Sx | Sum of desired elements with character A and character B. |
| Sn | Sum of all existing elements with character A and character B. |
| Calculates the expectation value of the hyper-geometric distribution. | |
| FN Hypergeo_D#(X,N,Sx,Sn) | |
| X | Desired number of elements with character A. |
| N | Number of elements with character A. |
| Sx | Sum of desired elements with character A and character B. |
| Sn | Sum of all existing elements with character A and character B. |
| Calculates the probability density of the hyper-geometric distribution. | |
| FN Poisson#(X,Lambda#) | |
| X | Variable. |
| Lambda# | Mean (is identical to variance in case of Poisson distribution). |
| Calculates the expectation value of the Poisson distribution. | |
| FN Poisson_D#(X,Lambda#) | |
| X | Variable. |
| Lambda# | Mean (is identical to variance in case of Poisson distribution). |
| Calculates the probability density of the Poisson distribution. | |
| 2.4 Some Random Number Generators |
| Random number generators are important aids for simulations. The Statistic Library offers you the
following random number generators: standard normal distributed, normal distributed, chi square and Fisher distributed.
If you need other random number generators, just call the inverse of the required distribution using an RND(0) instruction. For example: the function defined with FN Rnd_Expo#=FN Expo_Inv#(RND(0),5), yields the exponentially distributed random numbers with a mean of 5. |
| FN Rnd_Standard#(Dummy) | |
| Dummy | This parameter has no significance but has to be indicated nevertheless. |
| Calculates a standard normal distributed random number. | |
| FN Rnd_Normal#(Mu#,Var#) | |
| Mu# | Mean. |
| Var# | Variance. |
| Calculates a normal distributed random number. | |
| FN Rnd_Chi2#(I) | |
| I | Number of degrees of freedom. |
| Calculates a chi square distributed random number. | |
| FN Rnd_Fisher#(I1,I2) | |
| I1 | Number of degrees of freedom in numerator. |
| I2 | Number of degrees of freedom in denominator. |
| Calculates a Fisher distributed random number. | |
| 2.5 Confidence Intervals |
| As already discussed above, the functions in 2.2 always supply nothing more than the approximate value for the mean and the variance, respectively. Using the method of "maximum likelihood," it is possible to prove that these values are the best values for the parameter of the distribution function of the population, which spawned the data, but it is still only an estimate. Therefore, one has to indicate confidence limits (confidence intervals) for all basic statistics. The following procedures may be used for the most frequent distribution functions: |
| Conf_Mean_Normal_One Mean#,Var#,N,Alp#,R L#,R R#,Flag | |
| Mean# | Mean estimated from the measured values. |
| Var# | Variance of the population. |
| N | Size of sample. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | With the probability Alp#,the true mean is larger than this value. L#=0, if the calculation could not be performed. |
| R# | With the probability Alp#, the true mean is smaller than than this value. R#=0, if the calculation could not be performed. |
| Flag | In this parameter, a zero (0) has to be passed, if the variance was estimated from the sample. One (1) has to be passed if known from other information. |
| This procedure calculates the confidence interval for the mean with a normal distributed population and one-sided delimitation. This means that with a probability of Alp# , the mean is larger than L# or smaller than R#, respectively. | |
| Example: The example is a normal distributed sample with the mean Mean#=39.55, a previously known variance Var#=9, a size of N=10, and a confidence probability of Alp#=0.95. Since the variance is already known, we used Flag=1 to obtain the result: L#=37.99. Thus, the mean has a probability of 95% to be above 37.99. Stat_Init Conf_Mean_Normal_One 39.55,9,10,0.95,L#,R#,1 PRINT L# INPUT "End with [Return]";Dummy Stat_Exit END |
| Conf_Mean_Normal_Two Mean#,Var#,N,Alp#,R L#,R R#,Flag | |
| Mean# | Mean estimated from the measured values. |
| Var# | Variance of population. |
| N | Size of sample. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | Left limit of interval, encompassing the true mean with the probability of Alp#. L#=0, if the calculation could not be performed. |
| R# | Right limit of interval, encompassing the true mean with the probability of Alp#. R#=0, if the calculation could not be performed. |
| Flag | In this parameter, a zero (0) has to be passed, if the variance was estimated from the sample. One (1) has to be passed if known from other information. |
| This procedure calculates the confidence interval for the mean with a normal distributed population and two-sided delimitation. This means that with a probability of Alp# , the mean is located in the interval delimited by L# and R#. | |
| Example: As an example we chose a normal distributed sample with the mean Mean#=39.55, a previously known variance of Var#=9, a size of N=10, and a confidence probability of Alp#=0.95. The variance is already known, we receive the following results using Flag=1: L#=37.69 and R#=41.41. Thus, the mean has a 95% probability to be between 37.69 and 41.41. Stat_Init Conf_Mean_Normal_Two 39.55,9,10,0.95,L#,R#,1 PRINT L#,R# INPUT "End with [Return]";Dummy Stat_Exit END |
| Conf_Var_Normal_One Var#,N,Alp#,R L#,R R# | |
| Var# | Variance estimated from the measured values. |
| N | Size of sample. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | With a probability of Alp#, the true variance is larger than this value. L#=0, if the calculation could not be performed. |
| R# | With a probability of Alp#, the true variance is smaller than this value. R#=0, if the calculation could not be performed. |
| This procedure calculates the confidence interval for the variance with a normal distributed population and one-sided delimitation. This means that with a probability of Alp# , the variance is larger than L# or smaller than R#, respectively. | |
| Conf_Var_Normal_Two Var#,N,Alp#,R L#,R R# | |
| Var# | Variance estimated from the measured values. |
| N | Size of sample. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | Left limit of interval, encompassing the true variance with the probability of Alp#. L#=0, if the calculation could not be performed. |
| R# | Right limit of interval, encompassing the true variance with the probability of Alp#. R#=0, if the calculation could not be performed. |
| This procedure calculates the confidence interval for the variance with a normal distributed population and two-sided delimitation. This means that with a probability of Alp# , the mean is located in the interval delimited by L# and R#. | |
| Conf_Sigma_Normal_One S#,N,Alp#,R L#,R R# | |
| S# | Standard deviation estimated from the measured values. |
| N | Size of sample. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | With a probability of Alp#, the true standard deviation is larger than this value. L#=0, if the calculation could not be performed. |
| R# | With a probability of Alp#, the true standard deviation is smaller than this value. R#=0, if the calculation could not be performed. |
| This procedure calculates the confidence interval for the standard deviation with a normal distributed population and one-sided delimitation. This means that the standard deviation has the probability Alp# to be larger than L# or smaller than R#, respectively. | |
| Conf_Sigma_Normal_Two S#,N,Alp#,R L#,R R# | |
| S# | Standard deviation estimated from the measured values. |
| N | Size of sample. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | Left limit of interval, encompassing the true standard deviation with the probability of Alp#. L#=0, if the calculation could not be performed. |
| R# | Right limit of interval, encompassing the true standard deviation with the probability of Alp#. R#=0, if the calculation could not be performed. |
| This procedure calculates the confidence interval for the standard deviation with a normal distributed population and two-sided delimitation. This means that with a probability of Alp# , the standard deviation is located in the interval delimited by L# and R#. | |
| Conf_Bin_P_Two X,N,Alp#,R L#,R R# | |
| X | Number of elements with character A estimated from the measured values. |
| N | Size of sample. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | Left limit of interval, encompassing the true true number of elements with character A with the probability of Alp#. L#=0, if the calculation could not be performed. |
| R# | Right limit of interval, encompassing the true number of elements with character A with the probability of Alp#. R#=0, if the calculation could not be performed. |
| This procedure calculates the confidence interval for the number of elements with character A with a binomial distributed population and two-sided delimitation. This means that the number of elements with character A have the probability Alp# to be located in the interval delimited by L# and R#. The probability of success P# is linked with X through the formula P#=X/N. | |
| Conf_Poisson_Lambda_Two X#,Alp#,R L#,R R# | |
| X# | Mean (=variance) estimated from the measured values. |
| Alp# | Accuracy of estimate (0.8<=Alp#<=1). |
| L# | Left limit of interval, encompassing the true mean (=variance) with the probability of Alp#. L#=0, if the calculation could not be performed. |
| R# | Right limit of interval, encompassing the true mean (=variance) with the probability of Alp#. R#=0, if the calculation could not be performed. |
| This procedure calculates the confidence interval for the mean with a Poisson distributed population and two-sided delimitation. This means that the mean (=variance) has the probability Alp# to be located in the interval delimited by L# and R#. | |
| 2.6 Testing Agreement with Specified Nominal Values | ||||||||||||||||||||
| The procedures described in this chapter serve to check whether the nominal values are met. | ||||||||||||||||||||
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| Test_Normal_Mu0_Two Mu0#,Mean#,Var#,N,Alp#,R Res,Flag | |
| Mu0# | Nominal value for the mean. |
| Mean# | Mean of the sample. |
| Var# | Variance of the sample. |
| N | Size of sample. |
| Alp# | Confidence probability. |
| Res | Results in 1, if the null hypothesis is true, i.e., if the mean deviates statistically significant from Mu0#; otherwise 0. |
| Flag | If the variance was estimated from the sample, 0 has to be passed. If known from other information, 1 has to be passed. |
| Under the assumption of a normal distributed population, this procedure tests whether the nominal value for the mean is adhered to within the framework of the confidence probability. | |
| Test_Normal_Var_One Var0#,Var#,N,Alp#,R Res1,R Res2 | |
| Var0# | Nominal value for the variance. |
| Var# | Variance of the sample. |
| N | Size of sample. |
| Alp# | Confidence probability. |
| Res1 | Results in 1, if values fell below the nominal value; otherwise 0. |
| Res2 | Results 1, if the nominal value was exceeded; otherwise 0. |
| Under the assumption of a normal distributed population, this procedure tests whether a nominal value for the variance fell below or was exceeded within the framework of the confidence probability. | |
| Test_Normal_Var_Two Var0#,Var#,N,Alp#,R Res | |
| Var0# | Nominal value for the variance. |
| Var# | Variance of the sample. |
| N | Size of sample. |
| Alp# | Confidence probability. |
| Res | Results in 1, if the null hypothesis is true, i.e., if the variance is unequal the nominal value; otherwise 0. |
| Under the assumption of a normal distributed population, this procedure tests whether the variance of the sample is equal the nominal value of the variance within the framework of the confidence probability. | |
| Test_Bin_P0_Two P0#,X,N,Alp#,R Res | |
| P0# | Nominal value for the probability. |
| X | Number of elements with character A. |
| N | Size of sample. |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| Res | Results in 1, if the null hypotheses is true, i.e., if the probability is unequal the nominal value; otherwise 0. if the test could not be performed, then Res=-1. |
| Under the assumption of a binomial distributed population, this procedure tests whether the probability of the sample equals the nominal value for the probability within the framework of the confidence probability. | |
| 2.7 Comparing Two Samples |
| When comparing two samples, it is best not to compare the measured values directly when using known distribution laws but rather to compare the parameters of the distributions. We have implemented procedures for normal and binomial populations to be used for comparisons. |
| Cmp_Normal_Mean_Two Mean1#,Var1#,N1,Mean2#,Var2#, N2,Alp#,Flag,R Res |
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| Mean1# | Mean of the first sample. |
| Var1# | Variance of the first sample. |
| N1 | Size of the first sample. |
| Mean2# | Mean of the second sample. |
| Var2# | Variance of the second sample. |
| N2 | Size of the second sample. |
| Alp# | Confidence probability. |
| Flag | You may pass 3 different values in Flag with
the following significance: Flag=1 : The variances are known from other sources. Flag=2 : The variances are unknown but equal. Flag=3 : The variances are unknown and different. |
| Res | If the mean of both of the normally distributed populations match, Res=0; otherwise Res=1. |
| This procedure checks whether the mean of two normally distributed populations match. | |
| Cmp_Normal_Var Var1#,N1,Var2#,N2,Alp#,R Res | |
| Var1# | Variance of the first sample. |
| N1 | Size of the first sample. |
| Var2# | Variance of the second sample. |
| N2 | Size of the second sample. |
| Alp# | Confidence probability. |
| Res | If the variances of both of the normally distributed populations match, then Res=0; otherwise Res=1. |
| This procedure checks whether the variances of two normally distributed populations match. | |
| Cmp_Binomial_P P1#,N1,P2#,N2,Alp#,R Res | |
| P1# | Probability of the first sample. |
| N1 | Size of the first sample. |
| P2# | Probability of the second sample. |
| N2 | Size of the second sample. |
| Alp# | Confidence probability. |
| Res | If the relative frequencies of both of the binomially distributed populations match, then Res=0; otherwise Res=1. If the test could not be performed, then Res=-1. |
| Comparison of the relative frequencies of two binomially distributed populations with the fourfold
test. The comparison yields practical result only if the sample sizes are sufficiently large, that means if the following is valid: N1+N2>=20 AND N1*(P1#*N1+P2#*N2)/(N1+N2)>=5 AND N2*(P1#*N1+P2#*N2)/(N1+N2)>=5 |
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| U_Test &X#(),M,&Y#(),N,Alp#,Flag,R Res | |
| X#(1:M) | First sample. |
| M | Size of the first sample (M>=8). |
| Y#(1:N) | Second sample. |
| N | Size of the second sample (N>=8). |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| Flag | You may pass 2 different values in Flag with
the following significance: Flag=1 : One-sided test (Mean_X#<=Mean_Y# versus Mean_X#>Mean_Y#). Flag=2 : Two-sided test. |
| Res | If both samples originate from the same population, then Res=0; otherwise Res=1. If the test could not be performed, then Res=-1. |
| If information about the population does not exist, the so-called U-test can still yield rather acceptable results, whether the two samples originate from the same population or not. | |
| 2.8 Goodness of Fit Tests |
| The previous assumptions have always been that any information about the population from which the sample stems was available. In case the population is not known, it is possible to use the goodness of fit tests. The supposed probability function is passed to the following procedures, which in turn then check whether the samples stemmed from this function. The most important and in most cases completely sufficient tests for a discrete or continual distribution function are certainly contained in the Statistic Library. These procedures require classified data in the field X#(,). If only unclassified data exist, these have to be classified first. It was a quite conscious decision not to implement a procedure for this, because the classification of data depends too much on the procedure. |
| Fit_Uniform &X#(,),G,M,Alp#,R P#,R Res | |
| X#(1:G,0:1) | Contains the classified measured values. X#(1:G,0)= Class centers. X#(1:G,1)= Class frequencies. |
| G | Number of groups. |
| M | Total number of measured values. |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| P# | The probability that a group will occur. |
| Res | Res=1, if a goodness of fit to the uniform distribution
is not possible; otherwise Res=0. If an error has occurred (e.g., Alp#<0.8), then Res=-1. |
| This procedure performs the chi square goodness of fit test for an important special case, the uniform distribution. | |
| Example: Imagine a die being rolled 840 times. While doing this, the die shows the following points: 1 = 188 times 2 = 142 times 3 = 114 times 4 = 101 times 5 = 134 times 6 = 161 times A test is to determine whether one can assume that each number of points occur with the same probability P#=1/6. A simple BASIC program to test this hypothesis might be as follows: |
| Stat_Init -Die:DATA 1,188,2,142,3,114,4,101,5,134,6,161 DIM A#(6,1) RESTORE Die FOR I=1 TO 6 READ A#(I,0),A#(I,1) NEXT I Fit_Equal(&A#(,),6,840,0.95,P#,Result) IF Result=1 THEN PRINT "Goodness of fit NOT possible." ELSE PRINT "Goodness of fit using uniform" PRINT "distribution with P=";P#;" is possible." ENDIF INPUT "End with [Return]";Dummy Stat_Exit END |
| Fit_Binomial &X#(,),G,M,N,Mean#,Alp#,R P#,R Res | |
| X#(1:G,0:1) | Contains the classified measured values. X#(1:G,0)= Class centers. X#(1:G,1)= Class frequencies. |
| G | Number of groups. |
| M | Total number of measured values. |
| N | Parameters of the binomial distribution (number of independent repetitions). |
| Mean# | Mean of the binomial distribution. |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| P# | The probability that a group will ocurr. |
| Res | Res=1, if a goodness of fit to the binomial distribution
is not possible; otherwise Res=0. If an error has occurred (e.g., Alp#<0.8), then Res=-1. |
| Performs out a chi square goodness of fit test for a binomial distribution defined by N and Mean#. | |
| Fit_General &X#(,),G,M,Alp#,&FN Prob#(0),R Res | |
| X#(1:G,0:1) | Contains the classified measured values. X#(1:G,0)= Class centers. X#(1:G,1)= Class frequencies. |
| G | Number of groups. |
| M | Total number of measured values. |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| FN Prob#(0) | This is a probability function, which you have to define yourself and whose address has to be passed
to the procedure. The name of the function may be changed of course. Caution: It is absolutely necessary that a valid function pointer is passed; otherwise the system may experience serious crashes. |
| Res | Res=1, if a goodness of fit to the function defined by
you is not possible; otherwise Res=0. If an error has occurred (e.g., Alp#<0.8), then Res=-1. |
| This procedure performs the chi square goodness of fit test for a general discrete probability function
defined by you, which has to possess the following characteristics: The function has to be a double float type and take over a parameter, in which the class centers are passed. The result must be that it returns the corresponding probability density, i.e., it has to be defined at least for the sample range of X#(1:G,0). |
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| Kolmo_Smir_Normal &X#(,),Mean#,Var#,G,M,Alp#,Flag,R Res | |
| X#(1:G,0:1) | Contains the classified measured values. X#(1:G,0)= Class centers. X#(1:G,1)= Class frequencies. |
| Mean# | Mean of the binomial distribution. |
| Var# | Variance of the binomial distribution. |
| G | Number of groups. |
| M | Total number of measured values (M>=33). |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| Flag | If Flag=1, then the test will be significantly more accurate. However, only the values 0.8,0.85,0.9,0.95,0.99 are still permitted for Alp#. |
| Res | Res=1, if a goodness of fit to the function defined by
you is not possible; otherwise Res=0. If an error has occurred (e.g., Alp#<0.8), then Res=-1. |
| If it is possible to assume that a continual probability function exists, then the Kolmogoroff-Smirnoff goodness of fit test is used. This procedure tests the goodness of fit to a normal distribution. | |
| Kolmo_Smir_General &X#(,),G,M,Alp#,Flag,&FN Prob#(0),R Res | |
| X#(1:G,0:1) | Contains the classified measured values. X#(1:G,0)= Class centers. X#(1:G,1)= Class frequencies. |
| G | Number of groups. |
| M | Total number of measured values (M>=33). |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| Flag | If Flag=1, then the test will be significantly more accurate. However, only the values 0.8,0.85,0.9,0.95,0.99 are still permitted for Alp#. |
| FN Prob#(0) | This is a probability function, which you have to define yourself and whose address has to be passed
to the procedure. The name of the function may be changed of course. Caution: It is absolutely necessary that a valid function pointer is passed; otherwise the system may experience serious crashes. |
| Res | Res=1, if a goodness of fit to the function defined by
you is not possible; otherwise Res=0. If an error has occurred (e.g., Alp#<0.8), then Res=-1. |
| This procedure performs the Kolmogoroff-Smirnoff goodness of fit test for a general discrete probability
function defined by you, which has to have the following characteristics: The function has to be of double float type and take over a parameter, in which the class centers are passed. The result must be that it returns the corresponding probability density, i.e., it has to be defined at least for the sample range of X#(1:G,0). |
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| 2.9 Multifold Tables | ||||||||||||||||||||||||||||||||||||
| Unfortunately, the exact use of fourfold, K*2, or RC fold tables cannot be explained here. For further details, please consult the literature referenced in the bibliography in the appendix. | ||||||||||||||||||||||||||||||||||||
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| Brandt_Snedecor &X(,),K,Alp#,R Res | |
| X#(1:K,0:1) | X#(1:K,0)= Corresponding number of elements with character (+). X#(1:K,1)= Corresponding number of elements with character (-). |
| K | Number of samples. |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| Res | If the samples stem from the same population, then Res=0; otherwise Res=1. If the test could not be performed, then Res=-1. |
| This procedure performs the K*2 fold chi square test according to BRANDT and SNEDECOR. It is the result of the fourfold test, which considers the possibility that it might not always be practical or useful to examine only two samples for their respective homogeneity, but quite often rather has to examine K samples, which feature two characters (+) and (-). The K*2 fold schematic is represented in this procedure by the field X#(,). | |
| Example from SACHS: Please assume that a total of 80 patients have been treated in a course of therapy. Of this population, 40 patients have been treated only symptomatically (i.e., only the symptoms but none of the causes were treated). The other group with 40 patients received a standard dose of a new medication. The result of the treatment is expressed in the valence (occupation number) through the following 3*2 schematic (red sections): |
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| The objective is now to analyze on the 95% level whether the therapeutical results for both therapies were equal or whether they differed. For this evaluation, we only need the inner (red section) of the schematic, since the rest can be obtained through a simple summation: |
| Stat_Init -Patients:DATA 14,22,18,16,8,2 DIM X#(3,1) RESTORE Patients FOR I=1 TO 3 READ X#(I,0),X#(I,1) NEXT I Brandt_Snedecor(&X#(,),3,0.95,Result) IF Result THEN PRINT "Different results" ELSE PRINT "Equal results" ENDIF INPUT "End with [Return]";Dummy Stat_Exit END |
| Twoway_R_C &X(,),R,C,Alp#,R Res | |
| X(1:R,1:C) | X#(1:K,1)= Corresponding number of elements with character (1). X#(1:K,2)= Corresponding number of elements with character (2). ........ ........ X#(1:K,C)= Corresponding number of elements with character (C). |
| R | Number of samples. |
| C | Number of characters. |
| Alp# | Confidence probability (0.8<=Alp#<=1). |
| Res | If the samples stem from the same population, then Res=0, otherwise Res=1. If the test could not be performed, then Res=-1. |
| The R*C schematic is the expansion of the K*2 schematic to include R rows and C columns. Thus, we have not only two characters (+ and -) but C different characters. Otherwise, the calculation does not undergo any significant changes. | |
| Example: In order better to illustrate our point, we will take the example from the BRAND-SNEDECOR test, with the only difference that now a third group enters the picture, which has received twice the amount of the normal dosage (this example is also from the book by SACHS):
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