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# Why does my t-statistic have a negative sign?

The t statistic may be positive or negative depending on the direction of the difference between the means of two independent groups (unpaired t), the difference in pairs of means (paired t) or if a value is negative or positive (one sample t-test assuming a single variable has a zero mean).

The sign of the t-statistic not relevant if we are interested in two-tailed tests where the direction of difference is not important but if the direction is specified apriori then we need to remember the outputted p-value is the probability of observing a difference at least as extreme as that observed in the sample.

One-tailed tests may be of two forms both with the same null hypothesis (denoted by H0):

H0: mean 1 = mean 2 vs either HA: mean 1 < mean 2 or HA: mean 1 > mean 2

For example, in an unpaired t-test, if the second mean is found to be higher than the first mean, and this is expected apriori, the one-tailed p-value is half the outputted two-tailed p-value in SPSS. If, however, the direction of difference between the means is the opposite to that expected apriori then the one-tailed p-value equals 1-0.5*(two-tailed p-value).

The default among statistical packages performing tests is to report two-tailed p-values. Because the most commonly used test statistic distributions (standard normal, Student's t) are symmetric about zero, most one-tailed p-values can be derived from the two-tailed p-values. The example below is taken from this UCLA article.

Below, we have the output from a two-sample t-test in [the statistical package] Stata. The test is comparing the mean male score to the mean female score. The null hypothesis is that the difference in means is zero. The two-sided alternative is that the difference in means is not zero. There are two one-sided alternatives that one could opt to test instead: that the male score is higher than the female score (diff > 0) or that the female score is higher than the male score (diff < 0). In this instance, Stata presents results for all three alternatives. Under the headings Ha: diff < 0 and Ha: diff > 0 are the results for the one-tailed tests. In the middle, under the heading Ha: diff \$\$\ne\$\$ 0 (which means that the difference is not equal to 0), are the results for the two-tailed test.

Two-sample t test with equal variances

 Group Obs Mean Std. Error Std. Dev. Male 91 50.121 1.080 10.305 Female 109 54.991 0.779 8.134 Diff -4.870 1.304 8.134

Degrees of freedom: 198

H0: mean(male) - mean(female) = diff = 0

 Ha: diff < 0 diff \$\$\ne\$\$ 0 diff > 0 t = -3.7341 t = -3.7341 t = -3.7341 P < t = 0.0001 P > |t| = 0.0002 P > t = 0.9999

Note that the test statistic, -3.7341, is the same for all of these tests. The two-tailed p-value is P > |t|. This can be rewritten as P(>3.7341) + P(< -3.7341). Because the t-distribution is symmetric about zero, these two probabilities are equal: P > |t| = 2 * P(< -3.7341). Thus, we can see that the two-tailed p-value is twice the one-tailed p-value for the alternative hypothesis that (diff < 0). The other one-tailed alternative hypothesis has a p-value of P(>-3.7341) = 1-(P<-3.7341) = 1-0.0001 = 0.9999. So, depending on the direction of the one-tailed hypothesis, its p-value is either 0.5*(two-tailed p-value) or 1-0.5*(two-tailed p-value) if the test statistic symmetrically distributed about zero.

In this example, the two-tailed p-value suggests rejecting the null hypothesis of no difference. Had we opted for the one-tailed test of (diff > 0), we would fail to reject the null because of our choice of tails.

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