Case Study 3. A Longitudinal Analysis of Lung Function

The purpose of this assignment is to analyze the changes in forced expiratory volume at 1 second (FEV1) over time, and to determine whether trend is different in the treatment group. The dataset which will be analyzed if from the Childhood Asthma Management Program (CAMP) with children ages 5 - 12 years. It includes data from 1993 - 1995.

The variables in the dataset are as follows:

id: Participant ID
POSFEVPP: FEV1 after the administration of a bronchodilator (units = percent of predicted values)
TG: Treatment group, Treatment group: A=budesonide, B=nedocromil, C=placebo
visitc: Followup visit in months
GENDER: m=male, f=female
ETHNIC: w=white,b=black,h=hispanic,o=other
age_rz: Age in years at start of the study
hemog: Hemoglobin (g/dl)
wbc: White Blood Cell count (1000 cells/ul)
agehome: Age of current home (years)
anypet: Any pets, 1=Yes 2=No
woodstove: Used wood stove for heating/cooking, 1=Yes 2=No
dehumid: Use a dehumidifier, 1=Yes 2=No 3=DK
parent_smokes: Either Parent/partner smokes in home, 1=Yes 2=No
any_smokes: Anyone (including visitors) smokes in home, 1=Yes 2=No

Data Management & Recreating Variables

The first step would create dichotomous variables and create homogeneity between variables when analyzing in regression models. Names of some variables were changed for more clarity during interpretation.

Data Exploration

Here, various characteristics (demographics, treatment groups, etc.) are explored within the dataset.

a) Baseline data

Summary of baseline FEV1

This table shows the baseline data from the first visit which includes the mean, standard deviation (SD), minimum, and maximum for age at the start of the study, and the FEV1.

Mean FEV1	SD of FEV1	Minimum FEV1	Maximum FEV1
102.82	12.77	64	154

Statistics of FEV1

b) Exploring the participants in different treatment groups

Treatment Groups	Number	Percentage
Placebo	275	39.57
Budesonide	210	30.22
Nedocromil	210	30.22

c) Mean age of students at the start of the study

Mean Age
8.38

d) Gender distribution in the dataset

Gender	Frequency
Male	412
Female	283

e) Number of subjects according to ethnic groups

Ethnicity	Frequency
White	479
Black	89
Hispanic	68
Other	59

The most number of subjects are white (479), while blacks are 89, hispanics 68, and other ethnicities are 59.

f) Summary of minimum, maximum, mean, standard deviation, and standard error of the FEV1 mean differences

Minimum	Maximum	Mean	SD
-70	45	0.43	10.33

The minimum difference is -70, maximum is 45, mean 0.43, and standard deviation is 10.33.

g) Comparing the number of subjects at various visits

Visit	Number
Month0	695
Month2	664
Month4	673
Month12	674
Month16	666

h) Plots

Fig. 1. FEV1 distribution every month

The plots for FEV1 for each visit seem to be normally distributed, without any significant variation from each other. The peak is about 100% of predicted for each follow-up month.

Fig. 2. Spaghetti plot showing distributions of FEV1 during each visit

From the figure above, none of the treatment groups follow a specific pattern for the FEV1 values.

Fig. 3. FEV1 mean difference in each treatment group

The plot above shows no significant difference in mean differences within the 3 treatment groups.

Missingness of data

The wide format of the longitudinal dataset is used to explore the missingness of the data.

Missingness pattern

##     id TG GENDER ETHNIC age_rz anypet parent_smokes any_smokes woodstove
## 593  1  1      1      1      1      1             1          1         1
## 17   1  1      1      1      1      1             1          1         1
## 13   1  1      1      1      1      1             1          1         1
## 12   1  1      1      1      1      1             1          1         1
## 2    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 2    1  1      1      1      1      1             1          1         1
## 6    1  1      1      1      1      1             1          1         1
## 2    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 4    1  1      1      1      1      1             1          1         1
## 4    1  1      1      1      1      1             1          1         1
## 13   1  1      1      1      1      1             1          1         1
## 2    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 7    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 6    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 3    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         1
## 1    1  1      1      1      1      1             1          1         0
##      0  0      0      0      0      0             0          0         1
##     dehumid wbc hemog Month0 agehome Month12 Month4 Month16 Month2 diff    
## 593       1   1     1      1       1       1      1       1      1    1   0
## 17        1   1     1      1       1       1      1       1      0    1   1
## 13        1   1     1      1       1       1      1       0      1    0   2
## 12        1   1     1      1       1       1      0       1      1    1   1
## 2         1   1     1      1       1       1      0       1      0    1   2
## 1         1   1     1      1       1       0      1       1      1    1   1
## 2         1   1     1      1       1       0      1       1      0    1   2
## 6         1   1     1      1       1       0      1       0      1    0   3
## 2         1   1     1      1       1       0      1       0      0    0   4
## 1         1   1     1      1       1       0      0       1      0    1   3
## 4         1   1     1      1       1       0      0       0      1    0   4
## 4         1   1     1      1       1       0      0       0      0    0   5
## 13        1   1     1      1       0       1      1       1      1    1   1
## 2         1   1     1      1       0       1      0       1      1    1   2
## 1         1   1     1      1       0       0      1       1      1    1   2
## 7         1   1     1      0       1       1      1       1      1    0   2
## 1         1   1     1      0       1       1      1       1      0    0   3
## 6         1   1     0      1       1       1      1       1      1    1   1
## 1         1   1     0      1       1       1      0       0      0    0   5
## 1         1   1     0      1       1       0      1       0      0    0   5
## 3         1   0     1      1       1       1      1       1      1    1   1
## 1         1   0     1      0       1       1      1       1      0    0   4
## 1         0   1     1      1       1       1      1       1      1    1   1
## 1         1   1     1      1       1       1      1       1      1    1   1
##           1   4     8      9      16      22     26      31     32   40 190

The pink boxes indicate missign data, and it shows no pattern to the missingness.

Visualize plot for missing data

There are very few data are missing according to the above plot.

Scatter plot for missing data

There seems to be some missingness of data according to the pattern and the table above. No clear trend or pattern of missingness. It is assumed to be missing completely in random (MCAR).

Designing models

Several types of generalized estimating equation (GEEs) will be run. The model which best fits our data will be considered for our analysis.

a) Independence Correlation

term	estimate	conf.low	conf.high
(Intercept)	103.55	102.24	104.85
TGBudesonide	1.37	-0.72	3.46
TGNedocromil	-0.82	-2.78	1.15

Beta0: The mean FEV1 for individuals receiving the placebo is 103.55 (95% CI: 102.24, 104.85).
Beta1: The average difference in FEV1 comparing budesonide to placebo is 1.37 (95% CI: -0.72, 3.46).
Beta2: The average difference in FEV1 comparing nedocromil to placebo is -0.82 (95% CI: -2.78, 1.15).

b) Exchangeable Correlation

term	estimate	conf.low	conf.high
(Intercept)	103.75	102.45	105.05
TGBudesonide	1.12	-0.96	3.20
TGNedocromil	-1.01	-2.97	0.95

Beta0: The mean FEV1 for individuals receiving the placebo is 103.75 (95% CI: 102.45, 105.05).
Beta1: The average difference in FEV1 comparing budesonide to placebo is 1.12 (95% CI: -0.96, 3.20).
Beta2: The average difference in FEV1 comparing nedocromil to placebo is -1.01 (95% CI: -2.97, 0.95).

c) Autoregressive Correlation

term	estimate	conf.low	conf.high
(Intercept)	103.41	102.09	104.72
TGBudesonide	0.68	-1.40	2.76
TGNedocromil	-1.12	-3.15	0.90

Beta0: The mean FEV1 for individuals receiving the placebo is 103.41 (95% CI: 102.09, 104.72).
Beta1: The average difference in FEV1 comparing budesonide to placebo is 0.68 (95% CI: -1.40, 2.76).
Beta2: The average difference in FEV1 comparing nedocromil to placebo is -1.12 (95% CI: -3.15, 0.90).

d) Unstructured Correlation

term	estimate	conf.low	conf.high
(Intercept)	103.68	102.38	104.97
TGBudesonide	1.03	-1.04	3.10
TGNedocromil	-1.04	-3.00	0.92

Beta0: The mean FEV1 for individuals receiving the placebo is 103.68 (95% CI: 102.38, 104.97).
Beta1: The average difference in FEV1 comparing budesonide to placebo is 1.03 (95% CI: -1.04, 3.10).
Beta2: The average difference in FEV1 comparing nedocromil to placebo is -1.04 (95% CI: -3.00, 0.92).

Comparing & selecting a model for our analysis

Independence QIC

	x
QIC	530585.2
QICu	530567.6
Quasi Lik	-265280.8
CIC	11.8
params	3.0
QICC	530585.2

Exchangeable QIC

	x
QIC	530623.58
QICu	530623.53
Quasi Lik	-265308.76
CIC	3.03
params	3.00
QICC	530623.60

Autoregressive QIC

	x
QIC	531496.11
QICu	531495.84
Quasi Lik	-265744.92
CIC	3.14
params	3.00
QICC	531496.12

Unstructured QIC

	x
QIC	530644.76
QICu	530644.72
Quasi Lik	-265319.36
CIC	3.02
params	3.00
QICC	530644.86

Independence and exchangeable structures have the lowest QICs.

Correlation Structure

This will give an idea of the type of correlation the data follows.

	Month0	Month2	Month4	Month12	Month16
Month0	1.00	0.78	0.76	0.74	0.67
Month2	0.78	1.00	0.80	0.77	0.72
Month4	0.76	0.80	1.00	0.77	0.74
Month12	0.74	0.77	0.77	1.00	0.81
Month16	0.67	0.72	0.74	0.81	1.00

Though the correlation structure follows an autoregressive pattern, an exchangeable model is more suitable for our data.

Data Analysis

Simple GEE Models

FEV1 by Treatment Group

Term	Estimate	Lower Bound	Upper Bound	P-value
(Intercept)	103.75	102.45	105.05	0.00
TGBudesonide	1.12	-0.96	3.20	0.29
TGNedocromil	-1.01	-2.97	0.95	0.31

Plot to show FEV1 by time

The plot above confirms linearity of time.

Analysis of time

When time is analyzed as a whole

Term	Estimate	Lower Bound	Upper Bound	P-value
(Intercept)	103.84	102.94	104.74	0.00
visitc	-0.01	-0.05	0.03	0.68

When time is analysed separately

Term	Estimate	P-value	Lower Bound	Upper Bound
(Intercept)	102.66	0.00	101.30	104.02
TGBudesonide	1.37	0.20	-0.72	3.46
TGNedocromil	-0.82	0.41	-2.79	1.14
visitcMonth2	1.88	0.00	1.20	2.56
visitcMonth4	0.98	0.01	0.28	1.68
visitcMonth12	1.26	0.00	0.56	1.96
visitcMonth16	0.36	0.37	-0.43	1.16

There are differences in the estimates between when the month is analyzed as a whole and separately. The beta0 decreased from 104.98 to 102.66, and the coefficients for budesonide and nedocromil increased (became more positively associated) when month was analyzed separately.

Finalized model

This model includes more covariates to acount for other factors like demographic, blood and environmental characteristics. The process of backwards elimination is used to eliminate variables which are not statistically significant and had large p-values.

Term	Estimate	Lower Bound	Upper Bound	P-value
(Intercept)	93.58	83.21	103.96	0.00
visitc12	1.22	0.52	1.92	0.00
visitc16	0.44	-0.37	1.24	0.28
visitc2	1.94	1.29	2.60	0.00
visitc4	1.08	0.39	1.77	0.00
TGBudesonide	1.37	-0.76	3.51	0.21
TGNedocromil	-0.87	-2.87	1.12	0.39
GENDERFemale	0.33	-1.43	2.09	0.71
ETHNICBlack	-0.30	-3.08	2.48	0.83
ETHNICHispanic	-1.70	-4.56	1.16	0.24
ETHNICOther	0.11	-2.82	3.04	0.94
age_rz	-0.17	-0.60	0.26	0.43
hemog	0.59	-0.02	1.21	0.06
wbc	0.02	-0.02	0.06	0.26
anypet1	1.80	-0.08	3.68	0.06
woodstove1	-0.84	-4.26	2.58	0.63
dehumid1	1.11	-2.28	4.49	0.52
parent_smokes1	6.71	2.03	11.39	0.00
any_smokes1	-7.03	-11.54	-2.52	0.00

	x
QIC	494934.39
QICu	494934.06
Quasi Lik	-247448.03
CIC	19.16
params	19.00
QICC	494934.65

Confounding & Interaction

Confounding

Age

Term	Estimate	Lower Bound	Upper Bound	P-value
(Intercept)	104.98	101.43	108.53	0.00
TGBudesonide	1.15	-0.94	3.23	0.28
TGNedocromil	-1.01	-2.98	0.95	0.31
visitc	-0.01	-0.05	0.03	0.68
age_rz	-0.14	-0.55	0.27	0.50

Gender

Term	Estimate	Lower Bound	Upper Bound	P-value
(Intercept)	103.95	102.37	105.54	0.00
TGBudesonide	1.14	-0.94	3.23	0.28
TGNedocromil	-0.97	-2.95	1.01	0.34
visitc	-0.01	-0.05	0.03	0.69
GENDERm	-0.28	-1.99	1.44	0.75

Race

Term	Estimate	Lower Bound	Upper Bound	P-value
(Intercept)	103.48	100.80	106.17	0.00
TGBudesonide	1.12	-0.96	3.20	0.29
TGNedocromil	-0.98	-2.95	0.98	0.33
visitc	-0.01	-0.05	0.03	0.69
ETHNICh	-0.87	-4.56	2.82	0.64
ETHNICo	0.87	-2.84	4.58	0.65
ETHNICw	0.47	-2.22	3.17	0.73

Interaction

This model is designed to observe any effect that the treatment group or the visit had on the outcome.

Term	Estimate	Lower Bound	Upper Bound
(Intercept)	103.97	102.60	105.35
visitc	-0.03	-0.11	0.04
TGBudesonide	0.64	-1.57	2.84
TGNedocromil	-1.09	-3.20	1.03
visitc:TGBudesonide	0.07	-0.03	0.17
visitc:TGNedocromil	0.01	-0.09	0.12

Assumptions

Normality

The histogram shows normal curve, confirming normality within our sample.

Homoscadasticity

Treatment Group

The boxplots do not have much variations from each other.

Follow-up Visit

For each of the visit, there are not many variations between the follow-up month,

Linear Mixed-Effects Model

Linear Mixed-Effects Model (LMM) can also be used to analyze the CAMP dataset. The LMM models are usually used for subject-specific data. Like GEE models, LMM models are also selected according to p-values of the coefficients, AIC and scientific expertise. After running LMM models, comparisons were made between the treatment groups in GEE and LMM models.

Both the LMM and GEE models show no significant differences within each other. Only variations are seen in the 95% CI range. The finalized GEE model seems to be the best fit for our dataset.