1. How to Analyze and Graphically Present Longitudinal Data
Ayumi Shintani, Ph.D., M.P.H.
Department of Biostatistics
For handouts and datasets:

2. Example 1. More than 2 repeated measures with 1 group
From Table 1 of Deal et al (1979): Role of respiratory heat exchange in production of exercise-induced asthma. J Appl Physiol 46:
Minute ventilation Volume vs. Temperature-Dry Gas Experiments
Ventilation in 1/min ID
-10 25 37 50 65 80
Mean SD
Slope 1
74.5
81.5
83.6
68.6
73.1
79.4
76.8
5.7
-0.01 2
75.5
84.6
70.6
87.3 73 75
77.7
6.7
-0.02 3
68.9
71.6
55.9
61.9
60.5
61.8
63.4
5.8
-0.40 4 57
61.3
54.1
59.2
56.6
58.8
57.8
2.5 5
78.3
84.9 64
62.2
60.1
78.7
71.4
10.5
-0.12 6 54
62.8 63 58 56
51.5
57.6
4.7
-0.04 7
72.5
68.3
67.8
71.5
67.7
68.8
2.8
-0.06 8
80.8
89.9
83.2 83
85.7
79.6
83.7
3.7
70.2
75.6
69.0
66.3
69.1
-4.5
9.8
11.0
11.1
10.3
10.8
4.6

3. ] Minute Ventilation Volume Temperature
Error Bars show 95.0% Cl of Mean
Dot/Lines show Means
Temp_10
Temp25
Temp37
Temp50
Temp65
Temp80
Temperature
60.0
70.0
80.0
Minute Ventilation Volume ]

4. Minute Ventilation Volume
90.0 ID 1 2 3 4
80.0 5 6 7 8
70.0
Minute Ventilation Volume
Dot/Lines show Means
60.0
50.0
Temperture -10
Temperture 37
Temperture 65
Temperture 25
Temperture 50
Temperture 80
Temperature

5. We want to analyze whether there is an association between minute ventilation volume vs. temperature. What’s hypothesis do you want to test, i.e., what exactly do you want to compare?
Null hypothesis 1: Mean of minute ventilation volume at different temperatures are the same.
Error Bars show 95.0% Cl of Mean
Dot/Lines show Means
Temp_10
Temp25
Temp37
Temp50
Temp65
Temp80
Temperature
60.0
70.0
80.0
Minute Ventilation Volume ]

6. First, let’s ignore repeated measures, perform one-way ANOVA
In order to perform ANOVA, you first need to transform data from (horizontal) to (longitudinal) format (longitudinal format uses only one variable for outcome measures as oppose to horizontal, where different outcome variable is created for each repeated measure).
In SPSS go: Data
Restructure
Step 1: Welcome Data Structure Wizard - Select the first choice on
Step 2: Variables to Cases: Number of variable groups - Select the first choice
Step 3: Variables to Cases: Select variables
* Select Temp_01 through Temp80 (in the right order) to Variables to be transposed box (use Shift key on your key board to highlight them all once),
* Type the name of the output variable (for example, Vent)
* Select ID to the Fixed variable box, Next
Step 4: Variables to Cases: Create Index Variables - Select the first choice on
Step 5: Variables to Cases: Create One Index Variable
Click on the first choice (sequential numbers)
Edit index variable from Index1 to TEMP (for temperature)
Step 6: Handling variables not selected
Keep and treat as fixed variables
The rest remains the same
Step 7: Do nothing
Finish (before you do this, make sure you saved the original horizontal file)
Recode the level of Temp to the actual values (-10, 25, 37, …80)
Save the new longitudinal dataset as Deal Longitudinal.sav

7. Now, let’s perform one-way ANOVA
Analyze
Compare Means
One-way ANOVA
Select Vent as dependent, Temp as Factor variables
Another way:
Analyze
General linear model
Univariate (Multivariate means when you have more than 1 dependent variables)
Select Change as dependent, Temp as fixed Factor variables
Click Plots
Select Temp as horizontal variable, Click ADD, Continue
Click Models
Select Custom
Select Temp into Model box, Continue OK
Author’s test for effect of temperature: One-way analysis of variances : F5,42=.72, p>0.5

8. Results of one-way ANOVA
Tests of Between-Subjects Effects
Dependent Variable: Vent a 5
82.773
.727
.607 1
.000 42 48 47
Source
Corrected Model
Intercept
Temp
Error
Total
Corrected Total
Type III Sum
of Squares df
Mean Square F
Sig.
R Squared = .080 (Adjusted R Squared = -.030) a.

9. The authors concluded that no differences existed between minute ventilation volume and temperatures. The flaw in this analysis are:
1 The normality of the distributions of ventilation volumes has not been checked.
Use Kruskal-Wallis test
2. The observations within a single subject are not independent. The subject identification was not used in the analysis. The analysis does not remove variation among subjects by considering each subject as his own control, making use of the fact that variation within subjects is usually less than the variation between subjects.

10. Linear mixed effect model (???)
Ignoring correlation among measurements
Considering correlation among measurements
Non-parametric
Kruskal-Wallis test
(p=0.613)
Friedman
Test
(p=0.023)
Parametric
One-way
ANOVA (GLM)
(p=0.607)
Linear mixed effect model
(???)
Problem 2
Problem 1

11. Are they independent from each other?
In order to solve the problem 2, we can use a linear mixed effect model. A linear mixed effect model is similar to linear regression (or general linear regression) where outcome variables are continuous. A linear regression assumes normality and independence of residuals. Similarly a linear mixed model requires normality assumption, however, it does not requires independence assumption. We will talk about normality part later, here let’s spend some time to learn about independence assumption. ID
temp
trans1 1
-10
74.5 25
81.5 37
83.6 50
68.6 65
73.1 80
79.4 2
75.5
84.6
70.6
87.3
73.0
75.0 3
68.9
71.6
55.9
Are they independent from each other?

12. A quick way to check this is to do correlation analysis.
Correlations
1.000
.952 **
.690
.786 *
.738
.929 .
.000
.058
.021
.037
.001 8
.667
.881
.071
.004
.762
.857
.833
.028
.007
.010
Correlation Coefficient
Sig. (2-tailed) N
Temperture -10
Temperture 25
Temperture 37
Temperture 50
Temperture 65
Temperture 80
Spearman's rho
Temperture
-10 25 37 50 65 80
Correlation is significant at the 0.01 level (2-tailed).
**.
Correlation is significant at the 0.05 level (2-tailed). *.
If each observation is independent, you would expect p>0.05 for Spearman
correlation coefficient.

13. In order to perform correlation analysis, data must be entered horizontally, you can use deal.sav dataset for this. In the case, you have created your original database longitudinally and want to convert it to horizontal, here is how to do it.
Read Deal.long.sav into SPSS
Before you restructure this data, recode negative value for Temperature variable, by either recoding it to (1,2,3,4,5,6) or replace -10 with 10.
Go to:
Data
Restructure
Step 1: Welcome Data Structure Wizard - Select the second choice “Restructure selected cases into variables, click “next”
Step 2: Select “PATIENT ID” to identifier variable box (upper left box)
“Temperature” to index variable box (lower left box), click “next” Step 3: Finish

14. Mathematical Presentation of Correlation Structures
Let rjk donates a correlation coefficient between the jth and kth repeated measures on the same patients.
R(rjk ) is the working correlation matrix of Y
This is what a linear regression model assumes.

15. Since observations are not independent for repeated measure data (observations within a patient are dependent), we cannot use the independence assumption. A linear mixed model requires assumption on “correlation structure”.
You need to make a guess on how measures taken repeatedly correlate.

16. Data on the above figure assumes variance of Y at each point is the same across all categories, correlation between any 2 sets of Ys are zero (independent). This structure is called independent (scaled identity in SPSS). This is what 2-way ANOVA assumes for a structure of error terms, which is now obvious not providing good fit to the isoproterenol data.

17. Let’s look at the next figure
Let’s look at the next figure. Variance of each Y are the same across all the categories, correlation between any 2 Ys are the same (ie, correlation is the same when 2 doses are closer or not). This structure is called Compound Symmetry (Exchangeable).

18. Let’s look at the next figure
Let’s look at the next figure. Variance of each Y are the same across all the categories, correlation between any 2 Ys is equal to r(distance between Ys) where -1<r<1 (ie, correlation is the same when 2 doses are closer or not). This structure is called First-Order-Autoregressive.

19. AR(1): Heterogeneous. This is a first-order autoregressive structure with heterogeneous variances.

20. Mathematical Presentation of Correlation Structures
Let rjk donates a correlation coefficient between the jth and kth repeated measures on the same patients.
R(rjk ) is the working correlation matrix of Y

21. Model for the correlation
Independence (called “Scaled Identity” in SPSS)
Correlation between any two observations within the same patient is independent

22. Model for the correlation (cont.)
Exchangeable (compound symmetry)
Any two distinct observations from the same patient have the same correlation coefficient ()

23. Model for the correlation (cont.)
Unstructured
Each jk has different value, no structure is assumed in R

24. Model for the correlation (cont.)
Auto regressive (1)
rjk is function of time lag between 2 points

25. Toeplitz: Often fits well for experimental data.
Toeplitz. This covariance structure has homogenous variances and heterogenous correlations between elements. The correlation between adjacent elements is homogenous across pairs of adjacent elements. The correlation between elements separated by a third is again homogenous, and so on.

26. Selection of correlation structure
If the number of repeats is small and data are balanced and complete, then an unstructured matrix is recommended
If observations are measured over time, then use a structure that accounts for correlation as function of time (i.e. auto-regressive), choose a model which provides the smallest AIC value.
If observations are clustered (i.e. no logical ordering) then exchangeable may be appropriate

27. The model is able to consider the following covariance structures for repeated
measures data available in SPSS.
Ante-Dependence: First Order
AR(1)
AR(1): Heterogeneous
ARMA(1,1)
Compound Symmetry
Compound Symmetry: Correlation Metric
Compound Symmetry: Heterogeneous
Diagonal
Factor Analytic: First Order
Factor Analytic: First Order, Heterogeneous
Huynh-Feldt
Scaled Identity
Toeplitz
Toeplitz: Heterogeneous
Unstructured
Unstructured: Correlations

28. Let’s use a linear mixed effect model to analyze Minute ventilation Volume data.
Read Isoproterenol.long2.sav into SPSS.
Analyze,
Mixed Models, Linear,
Select ID for Subject variable
Select Dose as Repeated variable
Select appropriate covariance structure in the Repeated Covariance Type
for example AR(1) heterogeneous, Continue
Dependent variable: Vent
Factor variable (categorical independent variable) : Temp
Covariates (continuous independent variables):
Click Fixed
Click Custom, highlight all independent variables in the box
Choose Main Effect, and put them (Temp) in the model box
Click EM Means
Select Temp into the “Display means for” box
Select Bonferroni method
Click Compare main effect, select reference category to “first”
Click Statistics
Choose Parameter estimates, tests for covariance parameters, covariance for residuals
Click Save
Select Residuals, Predicted Values OK

29. Result of the linear mixed Model with ARH(1)
Information Criteria a
-2 Restricted Log
Likelihood
Akaike's Information
Criterion (AIC)
Hurvich and Tsai's
Criterion (AICC)
Bozdogan's Criterion
(CAIC)
Schwarz's Bayesian
Criterion (BIC)
The information criteria are displayed in smaller-is-better
forms. a.
Dependent Variable: Minute Ventilation Volume.
Type III Tests of Fixed Effects a
Denominator
Source
Numerator df df F
Sig.
Intercept 1
7.049
.000
temp 5
21.222
3.099
.030 a.
Dependent Variable: Minute Ventilation Volume.
Pairwise Comparisons b
5.425
2.339
14.346
.178
-1.513
12.363
-2.413
3.412
16.480
1.000
7.516
-1.225
3.783
20.843
9.493
-3.938
3.843
23.946
6.813
-1.125
4.246
19.444
10.993
(J) Temperature
-10
(I) Temperature 25 37 50 65 80
Mean
Difference
(I-J)
Std. Error df
Sig. a
Lower Bound
Upper Bound
95% Confidence Interval for
Based on estimated marginal means
Adjustment for multiple comparisons: Bonferroni. a.
Dependent Variable: Minute Ventilation Volume. b.

30. P-values were not adjusted For multiple comparisons.
Pairwise Comparisons b
-5.425 *
2.339
14.346
.036
-.419
2.413
3.412
16.480
.489
-4.804
9.629
1.225
3.783
20.843
.749
-6.645
9.095
3.938
3.843
23.946
.316
-3.995
11.870
1.125
4.246
19.444
.794
-7.749
9.999
5.425
.419
10.431
7.838
2.678
16.722
.010
2.180
13.495
6.650
3.458
23.877
.066
-.489
13.789
9.363
3.787
26.581
.020
1.585
17.140
6.550
4.285
23.101
.140
-2.312
15.412
-2.413
-9.629
4.804
-7.838
-2.180
-1.188
2.748
19.753
.670
-6.925
4.550
1.525
3.528
21.550
-5.800
8.850
-1.288
4.177
22.832
.761
-9.932
7.357
-1.225
-9.095
6.645
-6.650
1.188
-4.550
6.925
2.713
2.579
17.391
.307
-2.720
8.145
-.100
3.513
20.601
.978
-7.415
7.215
-3.938
3.995
-9.363
-1.585
-1.525
-8.850
5.800
-2.713
-8.145
2.720
-2.813
2.496
14.017
.279
-8.165
2.540
-1.125
-9.999
7.749
-6.550
2.312
1.288
-7.357
9.932
.100
-7.215
7.415
2.813
-2.540
8.165
(J) Temperature 25 37 50 65 80
-10
(I) Temperature
Mean
Difference
(I-J)
Std. Error df
Sig. a
Lower Bound
Upper Bound
95% Confidence Interval for
Based on estimated marginal means
The mean difference is significant at the .05 level. *.
Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments). a.
Dependent Variable: Minute Vantilation Volume. b.
P-values were not adjusted
For multiple comparisons.
Select Bonferroni option to adjust for multiple comparisons.
Therefore,
With the adjustment,
None of the pair-wise
Analysis was significant.

31. Predicted model by the linear mixed effect model with ARH(1)
Correlation structure.

32. This value is bigger than one with ARH(1) on page 29
Performing One-way ANOVA using a linear mixed effect model option in SPSS
When you use independence (scaled identity) structure for correlation, the model becomes equivalent with one-way ANOVA.
Result of the linear mixed model ignoring dependency among repeated measures.
This value is bigger than one with ARH(1)
on page 29
Information Criteria a
-2 Restricted Log
Likelihood
Type III Tests of Fixed Effects a 1 42
.000 5
.727
.607
Source
Intercept
temp
Numerator df
Denominator df F
Sig.
Dependent Variable: Minute Ventilation Volume. a.
Akaike's Information
Criterion (AIC)
Hurvich and Tsai's
Criterion (AICC)
Bozdogan's Criterion
(CAIC)
Schwarz's Bayesian
Criterion (BIC)
The information criteria are displayed in smaller-is-better
forms. a.
Dependent Variable: Minute Ventilation Volume.

33. Predicted model by the linear mixed effect model with independence
Correlation structure.
Model parameter estimates are the same as those of the model with ARH(1)
however, standard errors are much smaller with a model with considering
correlation among repeated measures.

34. Performing residual diagnosis for a linear mixed model in SPSS.
After you perform the analysis on page 28, by using SAVE option,
you created 2 new variables; residuals and predicted values.
Go to graphics, histogram to create graph 1.
Go to graphics, Scatter plot to create graph 2.
Graph 2: Checking for trend
in residuals (if there is a trend,
you may want to try transformation of Y
Graph 1:Checking for normality

35. Fitting slopes (null hypothesis 2: slope = 0)
In the previous analysis, we treated temperature as a categorical variable, which assesses that means ventilation volume were the same or not. All pair-wise analysis did not show any difference due to power loss (by adjustment for multiple comparisons). In order to assess whether there is increasing or decreasing trend by temperature, we can off course analyze this data with regression slope by treating temperature as continuous instead of categorical.
LOWSS curve

36. Performing a linear mixed effect model to assess slope is greater or less than zero.
Read deal.long.sav into SPSS.
Analyze,
Mixed Models,
Linear,
Select ID for Subject variable
Select Temp as Repeated variable
Select appropriate covariance structure in the Repeated Covariance Type
for example AR(1) heterogeneous
Continue
Dependent variable: Vent
Factor variable (categorical independent variable) :
Covariates (continuous independent variables): Temp
Click Fixed
Click Custom, highlight all independent variables in the box
Choose Main Effect, and put them (Temp) in the model box
Click Statistics
Choose Parameter estimates, tests for covariance parameters
Click Save
Select Residuals, Predicted Values OK

37. Result of the linear mixed model with AHR(1) with temp as continuous.
Estimates of Fixed Effects a
8.409
22.861
.000
22.702
.306
.762
Parameter
Intercept
temp
Estimate
Std. Error df t
Sig.
Lower Bound
Upper Bound
95% Confidence Interval
Dependent Variable: Minute Ventilation Volume. a.
P=0.762 indicates that the slope is not different from zero.

38. A much simpler way to analyze this data:
Response feature analysis, i.e., analysis of summary measures: Using slope as a summary measure:
Read Deal Longitudianl.sav dataset into SPSS
Graphs, Interactive, Scatterplot,
Select Vent as Y-axis, Temp (as Scale) as X-axis, ID (as Categorical) as panel variable
Click Fit
Select “Include constant in equation” Prediction line for “Mean”
Fit line for ‘Total” OK

39. Then perform One-sample non-parametric test (N is small) for the slope
Now, open Deal.sav and create a new variable Slope and type each person’s slope value
Then perform One-sample non-parametric test (N is small) for the slope
In order to perform one-sample non-parametric
Test in SPSS, you need this trick.
Create a dummy variable with all 0’s
Then go:
Analyze
2-related samples
Select Slope and Dummy to test pair list
Select Wilcoxon as test type OK
(SPSS does not work with only SLOPE variable) ID
Slope
Dummy 1
-0.01 2
-0.02 3
-0.4 4 5
-0.12 6
-0.04 7
-0.06 8
Using slopes to test for trends:
Wilcoxon signed-rank tests: P=0.018
We now can conclude that there is a significant association between minute ventilation volume and temperature (as temperature increases, ventilation volume decreases)

40. Using summary measures can provide more intuitive and simplified approach which some times provides bigger power to detect differences.