Hypothesis Test

t-test = sample size < 30, t distribution
z-test = sample size > 30, z distribution

For multi-level regression, t-test is different as needed to handle 2 level.

• Level 1 = degree of freedom = M _{^{(total number of all observation)}} – r _{^{(total number of independent variable)}} – 1
• level 2 = degree of freedom = N – q – 1

Example
Model 1 = grand-mean-centered IQ variable along with the group mean

Model 2 = with-in group deviation variable

Only Y₀₁ is different between model

Between group regression
Model 1 = y₁₀ + y₀₁ = 2.265 + 0.647 = 2.912
Model 2 = y₁₀ + y₀₁ = 0 + 2.912 = 2.912
Model is equivalent representation of the data, only differ in the parametrization.

t-ratio
Model 1 = 0.647/0.264 = 2.45
2.45 != 0, within and between group regression coefficient are indeed different

t-ratio (within group)
Model 1 = 2.265/0.065 = 34.9
t-ratio (between group)
Model 2 = 2.912/0.262 = 11.1
Both test is significant, concluding positive within-group and between group, indeed different

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Confidence Interval

Val < 0.1 = look very close to normal bell-curve, good to use
0.1 < Val < 0.3 = can be use for rough estimation, not going to be perfect
0.3 < Val = messy

95% confidence interval for level-one variance = 40.43 ± 1.96 (0.96) = 40.43 ± 1.88
95% confidence interval for level-two variance = 8.68 ± 1.96 (1.1) = 8.68 ± 2.156