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scipy.stats.

# ttest_ind_from_stats

scipy.stats.ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2, equal_var=True, alternative='two-sided')

  T-test for means of two independent samples from descriptive statistics.

  This is a test for the null hypothesis that two independent samples have identical average (expected) values.

  Parameters:
    mean1 array_like

      The mean(s) of sample 1.

    std1 array_like

      The corrected sample standard deviation of sample 1 (i.e. `ddof=1`).

    nobs1 array_like

      The number(s) of observations of sample 1.

    mean2 array_like

      The mean(s) of sample 2.

    std2 array_like

      The corrected sample standard deviation of sample 2 (i.e. `ddof=1`).

    nobs2 array_like

      The number(s) of observations of sample 2.

    equal_var bool, optional

      If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]. If False, perform Welch’s t-test, which does not assume equal population variance [2].

    alternative{‘two-sided’, ‘less’, ‘greater’}, optional

      Defines the alternative hypothesis. The following options are available (default is ‘two-sided’):

      - ‘two-sided’: the means of the distributions are unequal.

      - ‘less’: the mean of the first distribution is less than the mean of the second distribution.

      - ‘greater’: the mean of the first distribution is greater than the mean of the second distribution.

      Added in version 1.6.0.

  Returns:
    statistic float or array

      The calculated t-statistics.

    pvalue float or array

      The two-tailed p-value.

  See also

  `scipy.stats.ttest_ind`

  Notes

  The statistic is calculated as `(mean1 - mean2)/se`, where `se` is the standard error. Therefore, the statistic will be positive when mean1 is greater than mean2 and negative when mean1 is less than mean2.

  This method does not check whether any of the elements of std1 or std2 are negative. If any elements of the std1 or std2 parameters are negative in a call to this method, this method will return the same result as if it were passed `numpy.abs(std1)` and `numpy.abs(std2)`, respectively, instead; no exceptions or warnings will be emitted.

  Array API Standard Support

  `ttest_ind_from_stats` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ⛔
  JAX | ✅ | ✅
  Dask | ✅ | n/a

  > See Support for the array API standard for more information.

  References

  [1]

  https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test

  [2]

  https://en.wikipedia.org/wiki/Welch%27s_t-test

  Examples

  Suppose we have the summary data for two samples, as follows (with the Sample Variance being the corrected sample variance):

                       Sample   Sample
                 Size   Mean   Variance
      Sample 1    13    15.0     87.5
      Sample 2    11    12.0     39.0

  Apply the t-test to this data (with the assumption that the population variances are equal):

      >>> import numpy as np
      >>> from scipy.stats import ttest_ind_from_stats
      >>> ttest_ind_from_stats(mean1=15.0, std1=np.sqrt(87.5), nobs1=13,
      ...                      mean2=12.0, std2=np.sqrt(39.0), nobs2=11)
      Ttest_indResult(statistic=0.9051358093310269, pvalue=0.3751996797581487)

  For comparison, here is the data from which those summary statistics were taken. With this data, we can compute the same result using `scipy.stats.ttest_ind`:

      >>> a = np.array([1, 3, 4, 6, 11, 13, 15, 19, 22, 24, 25, 26, 26])
      >>> b = np.array([2, 4, 6, 9, 11, 13, 14, 15, 18, 19, 21])
      >>> from scipy.stats import ttest_ind
      >>> ttest_ind(a, b)
      TtestResult(statistic=0.905135809331027,
                  pvalue=0.3751996797581486,
                  df=22.0)

  Suppose we instead have binary data and would like to apply a t-test to compare the proportion of 1s in two independent groups:

                        Number of    Sample     Sample
                  Size    ones        Mean     Variance
      Sample 1    150      30         0.2        0.161073
      Sample 2    200      45         0.225      0.175251

  The sample mean \(\hat{p}\) is the proportion of ones in the sample and the variance for a binary observation is estimated by \(\hat{p}(1-\hat{p})\).

      >>> ttest_ind_from_stats(mean1=0.2, std1=np.sqrt(0.161073), nobs1=150,
      ...                      mean2=0.225, std2=np.sqrt(0.175251), nobs2=200)
      Ttest_indResult(statistic=-0.5627187905196761, pvalue=0.5739887114209541)

  For comparison, we could compute the t statistic and p-value using arrays of 0s and 1s and scipy.stat.ttest_ind, as above.

      >>> group1 = np.array([1]*30 + [0]*(150-30))
      >>> group2 = np.array([1]*45 + [0]*(200-45))
      >>> ttest_ind(group1, group2)
      TtestResult(statistic=-0.5627179589855622,
                  pvalue=0.573989277115258,
                  df=348.0)