How to perform two-sample one-tailed t-test with numpy/scipy

When null hypothesis is Ho: P1>=P2 and alternative hypothesis is Ha: P1<P2. In order to test it in Python, you write ttest_ind(P2,P1). (Notice the position is P2 first).

first = np.random.normal(3,2,400)
second = np.random.normal(6,2,400)
stats.ttest_ind(first, second, axis=0, equal_var=True)

You will get the result like below
Ttest_indResult(statistic=-20.442436213923845,pvalue=5.0999336686332285e-75)

In Python, when statstic <0 your real p-value is actually real_pvalue = 1-output_pvalue/2= 1-5.0999336686332285e-75/2, which is approximately 0.99. As your p-value is larger than 0.05, you cannot reject the null hypothesis that 6>=3. when statstic >0, the real z score is actually equal to -statstic, the real p-value is equal to pvalue/2.

Ivc’s answer should be when (1-p/2) < alpha and t < 0, you can reject the less than hypothesis.

After trying to add some insights as comments to the accepted answer but not being able to properly write them down due to general restrictions upon comments, I decided to put my two cents in as a full answer.

First let’s formulate our investigative question properly. The data we are investigating is

A = np.array([0.19826790, 1.36836629, 1.37950911, 1.46951540, 1.48197798, 0.07532846])
B = np.array([0.6383447, 0.5271385, 1.7721380, 1.7817880])

with the sample means

A.mean() = 0.99549419
B.mean() = 1.1798523

I assume that since the mean of B is obviously greater than the mean of A, you would like to check if this result is statistically significant.

So we have the Null Hypothesis

H0: A >= B

that we would like to reject in favor of the Alternative Hypothesis

H1: B > A

Now when you call scipy.stats.ttest_ind(x, y), this makes a Hypothesis Test on the value of x.mean()-y.mean(), which means that in order to get positive values throughout the calculation (which simplifies all considerations) we have to call

stats.ttest_ind(B,A)

instead of stats.ttest_ind(B,A). We get as an answer

• t-value = 0.42210654140239207
• p-value = 0.68406235191764142

and since according to the documentation this is the output for a two-tailed t-test we must divide the p by 2 for our one-tailed test. So depending on the Significance Level alpha you have chosen you need

p/2 < alpha

in order to reject the Null Hypothesis H0. For alpha=0.05 this is clearly not the case so you cannot reject H0.

An alternative way to decide if you reject H0 without having to do any algebra on t or p is by looking at the t-value and comparing it with the critical t-value t_crit at the desired level of confidence (e.g. 95%) for the number of degrees of freedom df that applies to your problem. Since we have

df = sample_size_1 + sample_size_2 - 2 = 8

we get from a statistical table like this one that

t_crit(df=8, confidence_level=95%) = 1.860

We clearly have

t < t_crit

so we obtain again the same result, namely that we cannot reject H0.

Did you look at this:
How to calculate the statistics "t-test" with numpy

I think that is exactly what this questions is looking at.

Basically:

import scipy.stats
x = [1,2,3,4]
scipy.stats.ttest_1samp(x, 0)

Ttest_1sampResult(statistic=3.872983346207417, pvalue=0.030466291662170977)

is the same result as this example in R. https://stats.stackexchange.com/questions/51242/statistical-difference-from-zero

    from scipy.stats import ttest_ind  
    
    def t_test(x,y,alternative='both-sided'):
            _, double_p = ttest_ind(x,y,equal_var = False)
            if alternative == 'both-sided':
                pval = double_p
            elif alternative == 'greater':
                if np.mean(x) > np.mean(y):
                    pval = double_p/2.
                else:
                    pval = 1.0 - double_p/2.
            elif alternative == 'less':
                if np.mean(x) < np.mean(y):
                    pval = double_p/2.
                else:
                    pval = 1.0 - double_p/2.
            return pval

    A = [0.19826790, 1.36836629, 1.37950911, 1.46951540, 1.48197798, 0.07532846]
    B = [0.6383447, 0.5271385, 1.7721380, 1.7817880]

    print(t_test(A,B,alternative='greater'))
    0.6555098817758839

Based on this function from R: https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/t.test

def ttest(a, b, axis=0, equal_var=True, nan_policy='propagate',
          alternative='two.sided'):        
    tval, pval = ttest_ind(a=a, b=b, axis=axis, equal_var=equal_var,
                           nan_policy=nan_policy)
    if alternative == 'greater':
        if tval < 0:
            pval = 1 - pval / 2
        else:
            pval = pval / 2
    elif alternative == 'less':
        if tval < 0:
            pval /= 2
        else:
            pval = 1 - pval / 2
    else:
        assert alternative == 'two.sided'
    return tval, pval