# t value greater than 1.96 but p value greater than 0.05

When dealing with t statistics, we reject the null hypothesis (H0: β=0) if the t value is greater than 1.96 in absolute value with a level of significance of 0.05.

I have run this model, with the factor A with 2 levels and factor B with three levels:

ldepvar ~ A * B + (1 + A * B | pp) + (1 + B | stim)


The output is:

 REML criterion at convergence: 2581

Scaled residuals:
Min      1Q  Median      3Q     Max
-4.6403 -0.5976  0.0223  0.6184  3.6417

Random effects:
Groups    Name                    Variance  Std.Dev.  Corr
stim      (Intercept)             3.810e-04 0.0195190
B2-1                    2.474e-07 0.0004974 1.00
B3-2                    3.000e-05 0.0054774 1.00  1.00
pp       (Intercept)             1.231e-02 0.1109433
A2-1                    3.389e-04 0.0184102  0.53
B2-1                    1.048e-02 0.1023753  0.10 -0.20
B3-2                    6.177e-03 0.0785929 -0.05  0.56 -0.57
A2-1:B2-1               5.700e-03 0.0754994 -0.13  0.20  0.44 -0.45
A2-1:B3-2               1.413e-03 0.0375945 -0.61  0.02 -0.42  0.80 -0.41
Residual                         9.012e-02 0.3001956
Number of obs: 5349, groups:  stim, 108; pp, 42

Fixed effects:
Estimate Std. Error         df t value Pr(>|t|)
(Intercept)         5.547963   0.017739  41.500000 312.756  < 2e-16       ***
A2-1                0.031541   0.008751 124.160000   3.604 0.000451 ***
B2-1               -0.153102   0.018941  38.630000  -8.083 7.86e-10 ***
B3-2                0.152832   0.015827  40.620000   9.657 4.46e-12 ***
A2-1:B2-1           0.047001   0.023612  36.540000   1.991 0.054048 .
A2-1:B3-2          -0.068961   0.020809 177.290000  -3.314 0.001115 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) A2-1 B2-1 B3-2 A2-1:2
A2-1      0.170
B2-1      0.079 -0.048
B3-2     -0.037  0.149 -0.538
A2-1:2-1 -0.057  0.013  0.185 -0.178
A2-1:3-2 -0.162 -0.008 -0.105  0.194 -0.470


If I wouldn't have the p values, I would have said that the interaction A2-1:B2-1 was significant at 5% significance (alpha lower than 0.05) level, as t=1.991. However, according to the p value it is significant at an alpha level of 0.1, as it is 0.054048. I would like to understand better the reason of this discrepancy.

## migrated from stackoverflow.comSep 21 '17 at 15:27

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• At the 5% significance level, you can reject when z, from the standard normal distribution is greater than 1.96. The critical value for a t statistics also depends on the sample size. For instance, the critical value of a t test when n = 2 is qt(.975, 2-1) = 12.7062. – Benjamin Sep 21 '17 at 15:20
• "we reject the null hypothesis (H0: β=0) if the t value is greater than 1.96 in absolute value with a level of significance of 0.05" ... this is not an accurate assertion. – Glen_b Sep 21 '17 at 22:36
• Thanks Benjamin. I missed that point. For anyone else in the future looking at this, you can calculate when your t value is significant with abs(qt(significance level, degrees of freedom). For example, if we want a significance level of 0.05 (two-tailed) and we have a sample size (number of participants in a sample) of 42 : abs(qt(0.05/2, 42-1)) – dede Sep 25 '17 at 9:55