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Default value: true
If stats_numer is true, inference statistical functions
return their results in floating point numbers. If it is false,
results are given in symbolic and rational format.
This is the mean t-test. Argument x is a list or a column matrix
containing an one dimensional sample. It also performs an asymptotic test
based on the Central Limit Theorem if option 'asymptotic is
true.
Options:
'mean, default 0, is the mean value to be checked.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'dev, default 'unknown, this is the value of the standard deviation when it is
known; valid values are: 'unknown or a positive expression.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic, default false, indicates whether it performs an exact t-test or
an asymptotic one based on the Central Limit Theorem;
valid values are true and false.
The output of function test_mean is an inference_result Maxima object
showing the following results:
'mean_estimate: the sample mean.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the population mean.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: \(p\)-value of the test.
Examples:
Performs an exact t-test with unknown variance. The null hypothesis is \(H_0: mean=50\) against the one sided alternative \(H_1: mean<50\); according to the results, the \(p\)-value is too great, there are no evidence for rejecting \(H_0\).
(%i1) load("stats")$
(%i2) data: [78,64,35,45,45,75,43,74,42,42]$
(%i3) test_mean(data,'conflevel=0.9,'alternative='less,'mean=50);
| MEAN TEST
|
| mean_estimate = 54.3
|
| conf_level = 0.9
|
| conf_interval = [minf, 61.51314273502712]
|
(%o3) | method = Exact t-test. Unknown variance.
|
| hypotheses = H0: mean = 50 , H1: mean < 50
|
| statistic = .8244705235071678
|
| distribution = [student_t, 9]
|
| p_value = .7845100411786889
This time Maxima performs an asymptotic test, based on the Central Limit Theorem.
The null hypothesis is \(H_0: equal(mean, 50)\) against the two sided alternative \(H_1: not equal(mean, 50)\);
according to the results, the \(p\)-value is very small, \(H_0\) should be rejected in
favor of the alternative \(H_1\). Note that, as indicated by the Method component,
this procedure should be applied to large samples.
(%i1) load("stats")$
(%i2) test_mean([36,118,52,87,35,256,56,178,57,57,89,34,25,98,35,
98,41,45,198,54,79,63,35,45,44,75,42,75,45,45,
45,51,123,54,151],
'asymptotic=true,'mean=50);
| MEAN TEST
|
| mean_estimate = 74.88571428571429
|
| conf_level = 0.95
|
| conf_interval = [57.72848600856194, 92.04294256286663]
|
(%o2) | method = Large sample z-test. Unknown variance.
|
| hypotheses = H0: mean = 50 , H1: mean # 50
|
| statistic = 2.842831192874313
|
| distribution = [normal, 0, 1]
|
| p_value = .004471474652002261
This is the difference of means t-test for two samples.
Arguments x1 and x2 are lists or column matrices
containing two independent samples. In case of different unknown variances
(see options 'dev1, 'dev2 and 'varequal bellow),
the degrees of freedom are computed by means of the Welch approximation.
It also performs an asymptotic test
based on the Central Limit Theorem if option 'asymptotic is
set to true.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'dev1, default 'unknown, this is the value of the standard deviation
of the x1 sample when it is known; valid values are: 'unknown or a positive expression.
'dev2, default 'unknown, this is the value of the standard deviation
of the x2 sample when it is known; valid values are: 'unknown or a positive expression.
'varequal, default false, whether variances should be considered to be equal or not;
this option takes effect only when 'dev1 and/or 'dev2 are 'unknown.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic, default false, indicates whether it performs an exact t-test or
an asymptotic one based on the Central Limit Theorem;
valid values are true and false.
The output of function test_means_difference is an inference_result Maxima object
showing the following results:
'diff_estimate: the difference of means estimate.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the difference of means.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: \(p\)-value of the test.
Examples:
The equality of means is tested with two small samples x and y, against the alternative \(H_1: m_1>m_2\), being \(m_1\) and \(m_2\) the populations means; variances are unknown and supposed to be different.
(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_means_difference(x,y,'alternative='greater);
| DIFFERENCE OF MEANS TEST
|
| diff_estimate = 20.31999999999999
|
| conf_level = 0.95
|
| conf_interval = [- .04597417812882298, inf]
|
(%o4) | method = Exact t-test. Welch approx.
|
| hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
|
| statistic = 1.838004300728477
|
| distribution = [student_t, 8.62758740184604]
|
| p_value = .05032746527991905
The same test as before, but now variances are supposed to be equal.
(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: matrix([1.2],[6.9],[38.7],[20.4],[17.2])$
(%i4) test_means_difference(x,y,'alternative='greater,
'varequal=true);
| DIFFERENCE OF MEANS TEST
|
| diff_estimate = 20.31999999999999
|
| conf_level = 0.95
|
| conf_interval = [- .7722627696897568, inf]
|
(%o4) | method = Exact t-test. Unknown equal variances
|
| hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
|
| statistic = 1.765996124515009
|
| distribution = [student_t, 9]
|
| p_value = .05560320992529344
This is the variance chi^2-test. Argument x is a list or a column matrix containing an one dimensional sample taken from a normal population.
Options:
'mean, default 'unknown, is the population’s mean, when it is known.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'variance, default 1, this is the variance value (positive) to be checked.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
The output of function test_variance is an inference_result Maxima object
showing the following results:
'var_estimate: the sample variance.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the population variance.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter.
'p_value: \(p\)-value of the test.
Examples:
It is tested whether the variance of a population with unknown mean is equal to or greater than 200.
(%i1) load("stats")$
(%i2) x: [203,229,215,220,223,233,208,228,209]$
(%i3) test_variance(x,'alternative='greater,'variance=200);
| VARIANCE TEST
|
| var_estimate = 110.75
|
| conf_level = 0.95
|
| conf_interval = [57.13433376937479, inf]
|
(%o3) | method = Variance Chi-square test. Unknown mean.
|
| hypotheses = H0: var = 200 , H1: var > 200
|
| statistic = 4.43
|
| distribution = [chi2, 8]
|
| p_value = .8163948512777689
This is the variance ratio F-test for two normal populations. Arguments x1 and x2 are lists or column matrices containing two independent samples.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'mean1, default 'unknown, when it is known, this is the mean of
the population from which x1 was taken.
'mean2, default 'unknown, when it is known, this is the mean of
the population from which x2 was taken.
'conflevel, default 95/100, confidence level for the confidence interval of the
ratio; it must be an expression which takes a value in (0,1).
The output of function test_variance_ratio is an inference_result Maxima object
showing the following results:
'ratio_estimate: the sample variance ratio.
'conf_level: confidence level selected by the user.
'conf_interval: confidence interval for the variance ratio.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: \(p\)-value of the test.
Examples:
The equality of the variances of two normal populations is checked against the alternative that the first is greater than the second.
(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_variance_ratio(x,y,'alternative='greater);
| VARIANCE RATIO TEST
|
| ratio_estimate = 2.316933391522034
|
| conf_level = 0.95
|
| conf_interval = [.3703504689507268, inf]
|
(%o4) | method = Variance ratio F-test. Unknown means.
|
| hypotheses = H0: var1 = var2 , H1: var1 > var2
|
| statistic = 2.316933391522034
|
| distribution = [f, 5, 4]
|
| p_value = .2179269692254457
Inferences on a proportion. Argument x is the number of successes in n trials in a Bernoulli experiment with unknown probability.
Options:
'proportion, default 1/2, is the value of the proportion to be checked.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic, default false, indicates whether it performs an exact test
based on the binomial distribution, or an asymptotic one based on the Central Limit Theorem;
valid values are true and false.
'correct, default true, indicates whether Yates correction is applied or not.
The output of function test_proportion is an inference_result Maxima object
showing the following results:
'sample_proportion: the sample proportion.
'conf_level: confidence level selected by the user.
'conf_interval: Wilson confidence interval for the proportion.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: \(p\)-value of the test.
Examples:
Performs an exact test. The null hypothesis is \(H_0: p=1/2\) against the one sided alternative \(H_1: p<1/2\).
(%i1) load("stats")$
(%i2) test_proportion(45, 103, alternative = less);
| PROPORTION TEST
|
| sample_proportion = .4368932038834951
|
| conf_level = 0.95
|
| conf_interval = [0, 0.522714149150231]
|
(%o2) | method = Exact binomial test.
|
| hypotheses = H0: p = 0.5 , H1: p < 0.5
|
| statistic = 45
|
| distribution = [binomial, 103, 0.5]
|
| p_value = .1184509388901454
A two sided asymptotic test. Confidence level is 99/100.
(%i1) load("stats")$
(%i2) fpprintprec:7$
(%i3) test_proportion(45, 103,
conflevel = 99/100, asymptotic=true);
| PROPORTION TEST
|
| sample_proportion = .43689
|
| conf_level = 0.99
|
| conf_interval = [.31422, .56749]
|
(%o3) | method = Asympthotic test with Yates correction.
|
| hypotheses = H0: p = 0.5 , H1: p # 0.5
|
| statistic = .43689
|
| distribution = [normal, 0.5, .048872]
|
| p_value = .19662
Inferences on the difference of two proportions. Argument x1 is the number of successes in n1 trials in a Bernoulli experiment in the first population, and x2 and n2 are the corresponding values in the second population. Samples are independent and the test is asymptotic.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided (p1 # p2), 'greater (p1 > p2)
and 'less (p1 < p2).
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'correct, default true, indicates whether Yates correction is applied or not.
The output of function test_proportions_difference is an inference_result Maxima object
showing the following results:
'proportions: list with the two sample proportions.
'conf_level: confidence level selected by the user.
'conf_interval: Confidence interval for the difference of proportions p1 - p2.
'method: inference procedure and warning message in case of any of the samples sizes
is less than 10.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: \(p\)-value of the test.
Examples:
A machine produced 10 defective articles in a batch of 250.
After some maintenance work, it produces 4 defective in a batch of 150.
In order to know if the machine has improved, we test the null
hypothesis H0:p1=p2, against the alternative H0:p1>p2,
where p1 and p2 are the probabilities for one produced
article to be defective before and after maintenance. According to
the p value, there is not enough evidence to accept the alternative.
(%i1) load("stats")$
(%i2) fpprintprec:7$
(%i3) test_proportions_difference(10, 250, 4, 150,
alternative = greater);
| DIFFERENCE OF PROPORTIONS TEST
|
| proportions = [0.04, .02666667]
|
| conf_level = 0.95
|
| conf_interval = [- .02172761, 1]
|
(%o3) | method = Asymptotic test. Yates correction.
|
| hypotheses = H0: p1 = p2 , H1: p1 > p2
|
| statistic = .01333333
|
| distribution = [normal, 0, .01898069]
|
| p_value = .2411936
Exact standard deviation of the asymptotic normal distribution when the data are unknown.
(%i1) load("stats")$
(%i2) stats_numer: false$
(%i3) sol: test_proportions_difference(x1,n1,x2,n2)$
(%i4) last(take_inference('distribution,sol));
1 1 x2 + x1
(-- + --) (x2 + x1) (1 - -------)
n2 n1 n2 + n1
(%o4) sqrt(---------------------------------)
n2 + n1
This is the non parametric sign test for the median of a continuous population. Argument x is a list or a column matrix containing an one dimensional sample.
Options:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
'median, default 0, is the median value to be checked.
The output of function test_sign is an inference_result Maxima object
showing the following results:
'med_estimate: the sample median.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: \(p\)-value of the test.
Examples:
Checks whether the population from which the sample was taken has median 6, against the alternative \(H_1: median > 6\).
(%i1) load("stats")$
(%i2) x: [2,0.1,7,1.8,4,2.3,5.6,7.4,5.1,6.1,6]$
(%i3) test_sign(x,'median=6,'alternative='greater);
| SIGN TEST
|
| med_estimate = 5.1
|
| method = Non parametric sign test.
|
(%o3) | hypotheses = H0: median = 6 , H1: median > 6
|
| statistic = 7
|
| distribution = [binomial, 10, 0.5]
|
| p_value = .05468749999999989
This is the Wilcoxon signed rank test to make inferences about the median of a continuous population. Argument x is a list or a column matrix containing an one dimensional sample. Performs normal approximation if the sample size is greater than 20, or if there are zeroes or ties.
See also pdf_rank_test and cdf_rank_test
Options:
'median, default 0, is the median value to be checked.
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
The output of function test_signed_rank is an inference_result Maxima object
with the following results:
'med_estimate: the sample median.
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameter(s).
'p_value: \(p\)-value of the test.
Examples:
Checks the null hypothesis \(H_0: median = 15\) against the alternative \(H_1: median > 15\). This is an exact test, since there are no ties.
(%i1) load("stats")$
(%i2) x: [17.1,15.9,13.7,13.4,15.5,17.6]$
(%i3) test_signed_rank(x,median=15,alternative=greater);
| SIGNED RANK TEST
|
| med_estimate = 15.7
|
| method = Exact test
|
(%o3) | hypotheses = H0: med = 15 , H1: med > 15
|
| statistic = 14
|
| distribution = [signed_rank, 6]
|
| p_value = 0.28125
Checks the null hypothesis \(H_0: equal(median, 2.5)\) against the alternative \(H_1: not equal(median, 2.5)\). This is an approximated test, since there are ties.
(%i1) load("stats")$
(%i2) y:[1.9,2.3,2.6,1.9,1.6,3.3,4.2,4,2.4,2.9,1.5,3,2.9,4.2,3.1]$
(%i3) test_signed_rank(y,median=2.5);
| SIGNED RANK TEST
|
| med_estimate = 2.9
|
| method = Asymptotic test. Ties
|
(%o3) | hypotheses = H0: med = 2.5 , H1: med # 2.5
|
| statistic = 76.5
|
| distribution = [normal, 60.5, 17.58195097251724]
|
| p_value = .3628097734643669
This is the Wilcoxon-Mann-Whitney test for comparing the medians of two continuous populations. The first two arguments x1 and x2 are lists or column matrices with the data of two independent samples. Performs normal approximation if any of the sample sizes is greater than 10, or if there are ties.
Option:
'alternative, default 'twosided, is the alternative hypothesis;
valid values are: 'twosided, 'greater and 'less.
The output of function test_rank_sum is an inference_result Maxima object
with the following results:
'method: inference procedure.
'hypotheses: null and alternative hypotheses to be tested.
'statistic: value of the sample statistic used for testing the null hypothesis.
'distribution: distribution of the sample statistic, together with its parameters.
'p_value: \(p\)-value of the test.
Examples:
Checks whether populations have similar medians. Samples sizes are small and an exact test is made.
(%i1) load("stats")$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) y:[21,18,25,14,52,65,40,43]$
(%i4) test_rank_sum(x,y);
| RANK SUM TEST
|
| method = Exact test
|
| hypotheses = H0: med1 = med2 , H1: med1 # med2
(%o4) |
| statistic = 22
|
| distribution = [rank_sum, 9, 8]
|
| p_value = .1995886466474702
Now, with greater samples and ties, the procedure makes normal approximation. The alternative hypothesis is \(H_1: median1 < median2\).
(%i1) load("stats")$
(%i2) x: [39,42,35,13,10,23,15,20,17,27]$
(%i3) y: [20,52,66,19,41,32,44,25,14,39,43,35,19,56,27,15]$
(%i4) test_rank_sum(x,y,'alternative='less);
| RANK SUM TEST
|
| method = Asymptotic test. Ties
|
| hypotheses = H0: med1 = med2 , H1: med1 < med2
(%o4) |
| statistic = 48.5
|
| distribution = [normal, 79.5, 18.95419580097078]
|
| p_value = .05096985666598441
Shapiro-Wilk test for normality. Argument x is a list of numbers, and sample
size must be greater than 2 and less or equal than 5000, otherwise, function
test_normality signals an error message.
Reference:
[1] Algorithm AS R94, Applied Statistics (1995), vol.44, no.4, 547-551
The output of function test_normality is an inference_result Maxima object
with the following results:
'statistic: value of the W statistic.
'p_value: \(p\)-value under normal assumption.
Examples:
Checks for the normality of a population, based on a sample of size 9.
(%i1) load("stats")$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) test_normality(x);
| SHAPIRO - WILK TEST
|
(%o3) | statistic = .9251055695162436
|
| p_value = .4361763918860381
Multivariate linear regression, \(y_i = b0 + b1*x_1i + b2*x_2i + ... + bk*x_ki + u_i\), where \(u_i\) are \(N(0,sigma)\) independent random variables. Argument x must be a matrix with more than one column. The last column is considered as the responses (\(y_i\)).
Option:
'conflevel, default 95/100, confidence level for the
confidence intervals; it must be an expression which takes a value
in (0,1).
The output of function linear_regression is an
inference_result Maxima object with the following results:
'b_estimation: regression coefficients estimates.
'b_covariances: covariance matrix of the regression
coefficients estimates.
b_conf_int: confidence intervals of the regression coefficients.
b_statistics: statistics for testing coefficient.
b_p_values: p-values for coefficient tests.
b_distribution: probability distribution for coefficient tests.
v_estimation: unbiased variance estimator.
v_conf_int: variance confidence interval.
v_distribution: probability distribution for variance test.
residuals: residuals.
adc: adjusted determination coefficient.
aic: Akaike’s information criterion.
bic: Bayes’s information criterion.
Only items 1, 4, 5, 6, 7, 8, 9 and 11 above, in this order,
are shown by default. The rest remain hidden until the user
makes use of functions items_inference and take_inference.
Example:
Fitting a linear model to a trivariate sample. The last column is considered as the responses (\(y_i\)).
(%i2) load("stats")$
(%i3) X:matrix(
[58,111,64],[84,131,78],[78,158,83],
[81,147,88],[82,121,89],[102,165,99],
[85,174,101],[102,169,102])$
(%i4) fpprintprec: 4$
(%i5) res: linear_regression(X);
| LINEAR REGRESSION MODEL
|
| b_estimation = [9.054, .5203, .2397]
|
| b_statistics = [.6051, 2.246, 1.74]
|
| b_p_values = [.5715, .07466, .1423]
|
(%o5) | b_distribution = [student_t, 5]
|
| v_estimation = 35.27
|
| v_conf_int = [13.74, 212.2]
|
| v_distribution = [chi2, 5]
|
| adc = .7922
(%i6) items_inference(res);
(%o6) [b_estimation, b_covariances, b_conf_int, b_statistics,
b_p_values, b_distribution, v_estimation, v_conf_int,
v_distribution, residuals, adc, aic, bic]
(%i7) take_inference('b_covariances, res);
[ 223.9 - 1.12 - .8532 ]
[ ]
(%o7) [ - 1.12 .05367 - .02305 ]
[ ]
[ - .8532 - .02305 .01898 ]
(%i8) take_inference('bic, res);
(%o8) 30.98
(%i9) load("draw")$
(%i10) draw2d(
points_joined = true,
grid = true,
points(take_inference('residuals, res)) )$