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| 73.1 Introduction to stats | ||
| 73.2 Functions and Variables for inference_result | ||
| 73.3 Functions and Variables for stats | ||
| 73.4 Functions and Variables for special distributions |
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Package stats contains a set of classical statistical inference and
hypothesis testing procedures.
All these functions return an inference_result Maxima object which contains
the necessary results for population inferences and decision making.
Global variable stats_numer controls whether results are given in
floating point or symbolic and rational format; its default value is true
and results are returned in floating point format.
Package descriptive contains some utilities to manipulate data structures
(lists and matrices); for example, to extract subsamples. It also contains some
examples on how to use package numericalio to read data from plain text
files. See descriptive and numericalio for more details.
Package stats loads packages descriptive, distrib and
inference_result.
For comments, bugs or suggestions, please contact the author at
'mario AT edu DOT xunta DOT es'.
@ref{Category: Statistical inference} · @ref{Category: Share packages} · @ref{Category: Package stats}
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Constructs an inference_result object of the type returned by the
stats functions. Argument title is a
string with the name of the procedure; values is a list with
elements of the form symbol = value and numbers is a list
with positive integer numbers ranging from one to length(values),
indicating which values will be shown by default.
Example:
This is a simple example showing results concerning a rectangle. The title of
this object is the string "Rectangle", it stores five results, named
'base, 'height, 'diagonal, 'area,
and 'perimeter, but only the first, second, fifth, and fourth
will be displayed. The 'diagonal is stored in this object, but it is
not displayed; to access its value, make use of function take_inference.
(%i1) load(inference_result)$
(%i2) b: 3$ h: 2$
(%i3) inference_result("Rectangle",
['base=b,
'height=h,
'diagonal=sqrt(b^2+h^2),
'area=b*h,
'perimeter=2*(b+h)],
[1,2,5,4] );
| Rectangle
|
| base = 3
|
(%o3) | height = 2
|
| perimeter = 10
|
| area = 6
(%i4) take_inference('diagonal,%);
(%o4) sqrt(13)
See also take_inference.
@ref{Category: Package stats}
Returns true or false, depending on whether obj is an
inference_result object or not.
@ref{Category: Package stats}
Returns a list with the names of the items stored in obj, which must
be an inference_result object.
Example:
The inference_result object stores two values, named 'pi and 'e,
but only the second is displayed. The items_inference function returns the names
of all items, no matter they are displayed or not.
(%i1) load(inference_result)$
(%i2) inference_result("Hi", ['pi=%pi,'e=%e],[2]);
| Hi
(%o2) |
| e = %e
(%i3) items_inference(%);
(%o3) [pi, e]
@ref{Category: Package stats}
Returns the n-th value stored in obj if n is a positive integer,
or the item named name if this is the name of an item. If the first
argument is a list of numbers and/or symbols, function take_inference returns
a list with the corresponding results.
Example:
Given an inference_result object, function take_inference is
called in order to extract some information stored in it.
(%i1) load(inference_result)$
(%i2) b: 3$ h: 2$
(%i3) sol: inference_result("Rectangle",
['base=b,
'height=h,
'diagonal=sqrt(b^2+h^2),
'area=b*h,
'perimeter=2*(b+h)],
[1,2,5,4] );
| Rectangle
|
| base = 3
|
(%o3) | height = 2
|
| perimeter = 10
|
| area = 6
(%i4) take_inference('base,sol);
(%o4) 3
(%i5) take_inference(5,sol);
(%o5) 10
(%i6) take_inference([1,'diagonal],sol);
(%o6) [3, sqrt(13)]
(%i7) take_inference(items_inference(sol),sol);
(%o7) [3, 2, sqrt(13), 6, 10]
See also inference_result and take_inference.
@ref{Category: Package stats}
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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.
@ref{Category: Package stats} · @ref{Category: Numerical evaluation}
This is the mean t-test. Argument x is a list or a column matrix
containing a 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
@ref{Category: Package stats}
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
@ref{Category: Package stats}
This is the variance chi^2-test. Argument x is a list or a column matrix containing a 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
@ref{Category: Package stats}
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
@ref{Category: Package stats}
This is the non parametric sign test for the median of a continuous population. Argument x is a list or a column matrix containing a 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
@ref{Category: Package stats}
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 a 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
@ref{Category: Package stats}
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
@ref{Category: Package stats}
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
@ref{Category: Package stats}
Simple linear regression, y_i=a+b x_i+e_i, where e_i are N(0,sigma) independent random variables. Argument x must be a two column matrix or a list of pairs.
Options:
'conflevel, default 95/100, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'regressor, default 'x, name of the independent variable.
The output of function simple_linear_regression is an inference_result Maxima object
with the following results:
'model: the fitted equation. Useful to make new predictions. See examples bellow.
'means: bivariate mean.
'variances: variances of both variables.
'correlation: correlation coefficient.
'adc: adjusted determination coefficient.
'a_estimation: estimation of parameter a.
'a_conf_int: confidence interval of parameter a.
'b_estimation: estimation of parameter b.
'b_conf_int: confidence interval of parameter b.
'hypotheses: null and alternative hypotheses about parameter b.
'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 about b.
'v_estimation: unbiased variance estimation, or residual variance.
'v_conf_int: variance confidence interval.
'cond_mean_conf_int: confidence interval for the conditioned mean. See examples bellow.
'new_pred_conf_int: confidence interval for a new prediction. See examples bellow.
'residuals: list of pairs (prediction, residual), ordered with respect to predictions.
This is useful for goodness of fit analysis. See examples bellow.
Only items 1, 4, 14, 9, 10, 11, 12, and 13 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 bivariate sample. Input %i4 plots
the sample together with the regression line; input %i5
computes y given x=113; the means and the
confidence interval for a new prediction when x=113 are also calculated.
(%i1) load("stats")$
(%i2) s:[[125,140.7], [130,155.1], [135,160.3], [140,167.2],
[145,169.8]]$
(%i3) z:simple_linear_regression(s,conflevel=0.99);
| SIMPLE LINEAR REGRESSION
|
| model = 1.405999999999985 x - 31.18999999999804
|
| correlation = .9611685255255155
|
| v_estimation = 13.57966666666665
|
(%o3) | b_conf_int = [.04469633662525263, 2.767303663374718]
|
| hypotheses = H0: b = 0 ,H1: b # 0
|
| statistic = 6.032686683658114
|
| distribution = [student_t, 3]
|
| p_value = 0.0038059549413203
(%i4) plot2d([[discrete, s], take_inference(model,z)],
[x,120,150],
[gnuplot_curve_styles, ["with points","with lines"]] )$
(%i5) take_inference(model,z), x=133;
(%o5) 155.808
(%i6) take_inference(means,z);
(%o6) [135.0, 158.62]
(%i7) take_inference(new_pred_conf_int,z), x=133;
(%o7) [132.0728595995113, 179.5431404004887]
@ref{Category: Package stats} · @ref{Category: Statistical estimation}
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Probability density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.
See also test_signed_rank.
@ref{Category: Package stats}
Cumulative density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.
See also test_signed_rank.
@ref{Category: Package stats}
Probability density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.
See also test_rank_sum.
@ref{Category: Package stats}
Cumulative density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.
See also test_rank_sum.
@ref{Category: Package stats}
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