1. From t-test to multilevel analyses Del-2
Stein Atle Lie, statistician, professor
Uni Health, Uni Research

2. Outline Pared t-test (Mean and standard deviation)
Two-group t-test (Mean and standard deviations)
Linear regression
GLM (general linear models)
GLMM (general linear mixed model) …
PASW (former SPSS), Stata, R, gllamm (Stata)

3. Pared t-test
The straightforward way to analyze two repeated measures is a pared t-test.
Measure at time1 or location1 (e.g. Data1) is directly compared to measure at time2 or location2 (e.g. Data2)
Is the difference between Data1 and Data2 (Diff = Data1-Data2) unlike 0?

4. Pared t-test
The pared t-test will only be performed for complete (balanced) data.
What happens if we delete two observations from data2?
(Only 8 complete pairs remain)

5. Two group t-test
If we now consider the data from time1 and time2 (or location1 and location2) to be independent (even if their not) and use a two group t-test on the full dataset, 2*10 observations

6. ID
Grp (or time)
Dummy
Data
1007 1
6,1
1008
4,6
1009 5
1010
4,8
1011
7,9
1012
13,8
1013
1,4
1014
9,5
1015
12,2
1016
8,5 2
8,3
9,8
15,6
9,2
10,4
3,3
8,4
16,1

7. Two group t-test (n=20 [10+10])
PASW:
T-TEST GROUPS=Grp(1 2)
/VARIABLES=Data.

8. Two group t-test
Observe that mean for Grp1 and Grp2 is equal to mean for Data1 and Data2 (from the pared t-test)
And that the mean difference is also equal!
The difference between pared t-test and two group t-test lies in the
Variance - and the number of observations
and therefore in the standard deviation and standard error
and hence in the p-value and confidence intervals

9. Two group t-test (s1=s2) s1 s2 m1 m2 D

10. Two group t-test (s1=s2) s1 s2

11. ANOVA (Analysis of variance (s1=s2=s3)m1 m2 m3

12. ANOVA (Analysis of variance (s1=s2=s3)

13. Linear regression
If we now perform an ordinary linear regression with the data as outcome (dependent variable) and the group (time) variable (Grp=1 and 2) as independent variable

14. Linear regression (n=20)
Stata:
. regress data grp
Source | SS df MS Number of obs =
F( 1, 18) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
data | Coef. Std. Err t P>|t| [95% Conf. Interval]
grp |
_cons |

15. Linear regression
The coefficient for group is identical to the mean difference
and the standard error, t-statistic, and p‑value are identical to those found in a two‑group t‑test

16. Linear regression
Now exchange the independent variable for group (Grp=1 and 2) with a dummy variable (dummy=0 for grp=1 and dummy=1 for grp=2)
the coefficient for the dummy is equal to the coefficient for grp (the mean difference)
and the coefficient for the constant term is equal to the mean for grp1 (the standard error is not!)

17. Linear regression (n=20)
Stata:
. regress data dummy
Source | SS df MS Number of obs =
F( 1, 18) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
data | Coef. Std. Err t P>|t| [95% Conf. Interval]
dummy |
_cons |

18. Linear models in Stata
In ordinary linear models (regress and glm) in Stata one may add an option for clustered data – to obtain robust standard errors adjusted for intragroup correlation.
(This is ideal when you want to adjust for clustered data, but are not interested in the correlation within or between groups)

19. Linear regression (n=20)
Stata:
. regress data dummy, cluster(id)
Linear regression Number of obs =
F( 1, 9) =
Prob > F =
R-squared =
Root MSE =
(Std. Err. adjusted for 10 clusters in id)
| Robust
data | Coef. Std. Err t P>|t| [95% Conf. Interval]
dummy |
_cons |

20. Linear models in Stata
Thus, we now have an alternative to the pared t‑test. The mean difference is identical to that obtained from the pared t‑test, and the standard errors (and p-values) are adjusted for intragroup correlation
As an alternative we may use the program gllamm (Generalized Linear Latent And Mixed Models) in Stata

21. gllamm (n=20) gllamm (Stata): . gllamm data dummy, i(id)
number of level 1 units = 20
number of level 2 units = 10
data | Coef. Std. Err z P>|z| [95% Conf. Interval]
dummy |
_cons |
Variance at level ( )
level 2 (id) var(1): ( )

22. Linear models in Stata
If we now delete two of the observations in Grp2
We then have coefficients (“mean differences”) calculated based on all (n=18) data
and standard errors corrected for intragroup correlation - using the commands <regress>, <glm> or <gllamm>

23. Linear regression (n=18)
Stata:
. regress data dummy, cluster(id)
Linear regression Number of obs =
F( 1, 9) =
Prob > F =
R-squared =
Root MSE =
(Std. Err. adjusted for 10 clusters in id)
| Robust
data | Coef. Std. Err t P>|t| [95% Conf. Interval]
dummy |
_cons |

24. gllamm (n=18) gllamm (Stata): . gllamm data dummy, i(id)
number of level 1 units = 18
number of level 2 units = 10
log likelihood =
data | Coef. Std. Err z P>|z| [95% Conf. Interval]
dummy |
_cons |
Variance at level ( )
level 2 (id) var(1): ( )

25. Intra class correlation (ICC)
Variance at level ( )
level 2 (id) var(1): ( )
The total variance is hence =
(and the standard deviation is hence )
The proportion of variance attributed to level 2 is therefore
ICC = / = 0.578

26. Linear regression Ordinary linear regression
Assumes data is Normal and i.i.d. (identical independent distributed)

27. Linear regression Y X Regression line: y = b0 + b1·x b1 b0
residual b1
(x1,y1)
(xn,yn)
(xi,yi) b0
Kortisol * Months
Height * Weight
Kortisol * Time X

28. Linear regression Assumptions:
1) y1, y2,…, yn are independent normal distributed
2) The expectation of Yi is: E(Yi) = b0 + b1·xi (linear relation between X and Y)
3) The variance of Yi is: var(Yi) = s2 (equal variance for ALL values of X)

29. Linear regression Assumptions - Residualer (ei): yi = a + b·xi + ei
1) e1, e2,…, en are independent normal distributed
2) The expectation of ei is: E(ei) = 0
3) The variance of ei is: var(Yi) = s2

30. Ordinary linear regression
The formula for an ordinary regression can thus be expressed as:
yi = b0 + b1·xi + ei
ei ~N(0, se2)

31. Random intercept modelY
Regression lines:
yij = b0 + b1·xij+vij
(x11,y11) b1
(xnp,ynp)
b0+uj
(xij,yij) su se X

32. Random intercept model
For a random intercept model, we can express the regression line(s) - and the variance components as
yij = b0 + b1·xij + vij
vij = uj + eij
eij ~N(0, se2) (individual)
uj ~N(0, su2) (group)

33. Random intercept model
Alternatively we may express the formulas, for the simple variance component model, in terms of random intercepts:
yij = b0j + b1·xij + eij
b0j = b0 + uj
eij ~N(0, se2) (individual)
uj ~N(0, su2) (group)