Basic statistical inference (mqr.inference)#

User guide

Basic Statistical Inference

Detailed examples

nklsxn/mqrpy-guide

This package contains wrappers and functions for basic inference on statistics calculated from samples of data. Many routines call either scipy or statsmodels (see function docs).

One of the main goals of this module is to organise statistical tests by purpose. The name of a test or method used in a routine is given in the method argument, instead of the function name.

Routines in the top level (mqr.inference) are parametric, while routines in the mqr.inference.nonparametric module are non-parametric.

The modules implement hypothesis tests for all of the listed statistics, and in addition, the parametric modules also calculate confidence intervals and sample-size.

Parametric statistics

dist

correlation

mean

proportion

rate

stddev

variance

Non-parametric statistics

nonparametric.correlation

nonparametric.dist

nonparametric.median

nonparametric.quantile

nonparametric.variance

Result types

ConfidenceInterval(name, method, value, ...)

Result type for confidence interval calculations.

HypothesisTest(description, alternative, ...)

Result of an hypothesis test.

TestPower(name, alpha, beta, effect, ...)

Result of a sample size calculation for an hypothesis test.

Notes#

NIST [1] has further information on how to compare processes and products using hypothesis tests.

Examples#

All functions have a similar interface and return one of the result types above.

This example calculates a confidence interval for the difference between Pearson correlation coefficients.

>>> x = np.array([1, 2, 1, 1, 2])
>>> y = np.array([3, 1, 2, 2, 1])
>>> mqr.inference.correlation.confint(x, y)
ConfidenceInterval(
    name='correlation',
    method='fisher-z',
    value=-0.8728715609439694,
    lower=-0.9915444093872248,
    upper=0.040865942695342376,
    conf=0.95,
    bounded='both')

This example calculates the sample size required to achieve an hypothesis test on the mean with standardised effect size (Cohen’s F) of 0.5, 95% confidence and 90% power.

>>> import mqr
>>> mqr.inference.mean.size_1sample(0.5, 0.05, 0.1, 'greater')
TestPower(
    name='mean',
    alpha=0.05,
    beta=0.1,
    effect=0.5,
    alternative='greater',
    method='t',
    sample_size=35.65267806948644)

Here is an example of an hypothesis test on the ratio of rates of events. Two samples were taken, one before a process improvement activity and another after. The sample before showed 5 defects in a sample of fabric 0.5m*0.5m, which is 20 defects per squqre metre. After the improvement, the same sized sample showed only 1 defect — that’s 4 defects per square metre on average. This tests the hypothesis that the defect rate has been decreased by at least 10 defects per square metre.

>>> import mqr
>>> mqr.inference.rate.test_2sample(
>>>     5, 0.25, 1, 0.25, H0_value=10,
>>>     alternative='less', compare='diff')
HypothesisTest(
    description='difference between rates of events',
    alternative='less',
    method='score',
    sample_stat='rate1 - rate2',
    sample_stat_target=10,
    sample_stat_value=16.0,
    stat=0.65209454185206,
    pvalue=0.7428299075363056,
    null='rate1 - rate2 == 10',
    alt='rate1 - rate2 < 10')

From these samples, there is not enough evidence to conclude that the defect rate has been improved by 10 defects per square metre.

This is an example of a test for normality. First, create a random sample of F-distributed data.

>>> import numpy as np
>>> import scipy.stats as st
>>> np.random.seed(0)
>>> data = st.f(1, 2).rvs(40)

Check the hypothesis that they are normally distributed using the Kolmogorov- Smirnov test.

>>> import mqr
>>> mqr.inference.dist.test_1sample(data, test='ks-norm')
HypothesisTest(
    description='non-normality',
    alternative='two-sided',
    method='kolmogorov-smirnov',
    sample_stat='distribution',
    sample_stat_target='normal',
    sample_stat_value=None,
    stat=0.2881073311024548,
    pvalue=0.0009999999999998899,
    null='distribution == normal',
    alt='distribution != normal')

References#

[1]

National Institute of Standards and Technology (US Dep. of Commerce). “Product and Process Comparisons.” https://www.itl.nist.gov/div898/handbook/prc/prc.htm

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