# Longitudinal analysis with intervention

I hope this question fits the topic of this forum.

Consider a dataset - a cohort of N subjects (s_i), with measurements at fixed time points (t_i) (every week) of some quantity (Q_ij) . Then after K weeks, some intervention happened, and we continued measuring the same quantity over the same cohort at the same time intervals (on a weekly basis) for another K weeks. In the matrix notations, the data set BEFORE and AFTER the intervention has the same form (there are the same amount of K weeks) and may be represented as:

     | t_1  | t_2  | t_3  |... | t_K  |
s_1| Q_11 | Q_12 | Q_13 |... | Q_1K |
s_2| Q_21 | Q_22 | Q_23 |... | Q_2K |
s_3| Q_31 | Q_32 | Q_33 |... | Q_3K |
.
.
.
s_N| Q_N1 | Q_N2 | Q_N3 |... | Q_NK |


QUESTION How to determine whether the intervention had any effect on the measured quantity Q ?

I am very new to statistics and longitudinal analysis, so this question may sound very basic.

1. Conceptually this is similar to paired t-test, but should be adjusted to the cohort size (ANOVA?)
2. Another approach is to compute a slope over the K points for each subject before and after the intervention and compare them, and then check if their difference is statistically significant based on errors and confidence intervals. And somehow to take into account all subjects (average over slopes?)

I am sure that is a standard problem in longitudinal analysis with standard approach to assess an affect of the intervention.

Any suggestions about toolboxes? I work with python/matlab, but R can be considered perfectly.

UPDATE

Here is the sample of my data: 20 subjects for 8 weeks of integer measurements at equally spaced times, before and after the intervention (rows correspond to the same subject before and after the intervention)

Before:

   0.00000    1.00000    1.00000    1.00000    1.00000    3.00000    3.00000    5.00000
16.00000   16.00000   16.00000   14.00000   12.00000   12.00000   12.00000   12.00000
3.00000    2.00000    3.00000   10.00000   10.00000   12.00000   14.00000   14.00000
5.00000    3.00000    2.00000    1.00000    0.00000    0.00000    1.00000    0.00000
10.00000    7.00000    3.00000    4.00000    3.00000    5.00000    4.00000    4.00000
8.00000    9.00000    9.00000    9.00000    6.00000    7.00000    8.00000   11.00000
5.00000    5.00000    3.00000    4.00000    8.00000    7.00000   11.00000    4.00000
3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    5.00000
10.00000   10.00000   10.00000   10.00000   10.00000   10.00000   10.00000   10.00000
17.00000   11.00000   12.00000   21.00000   18.00000   12.00000   15.00000   16.00000
7.00000    9.00000    8.00000    8.00000    7.00000    8.00000    7.00000    9.00000
13.00000   17.00000   14.00000   19.00000   20.00000   23.00000   23.00000   24.00000
3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    3.00000
7.00000    8.00000    4.00000    5.00000    5.00000   12.00000    9.00000   12.00000
15.00000   10.00000   13.00000   14.00000   12.00000   11.00000   13.00000   15.00000
2.00000    3.00000    2.00000    2.00000    2.00000    6.00000    2.00000    2.00000
3.00000    2.00000    3.00000    3.00000    1.00000    3.00000    3.00000    1.00000
3.00000    2.00000    3.00000    2.00000    1.00000    1.00000    4.00000    2.00000
13.00000   15.00000   13.00000    4.00000    7.00000    8.00000    9.00000    9.00000
0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000
8.00000    6.00000    7.00000   10.00000    3.00000    9.00000   10.00000    6.00000


After:

   3.00000    1.00000    3.00000    2.00000    1.00000    4.00000    3.00000    2.00000
12.00000    6.00000    6.00000    6.00000    6.00000    4.00000    3.00000    3.00000
15.00000   15.00000   12.00000    9.00000    1.00000    3.00000    3.00000    2.00000
0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000
2.00000    6.00000    3.00000    3.00000    6.00000    5.00000    5.00000    3.00000
9.00000    7.00000    9.00000    7.00000    8.00000    7.00000   10.00000    5.00000
8.00000    6.00000    6.00000    6.00000    6.00000    5.00000    3.00000    6.00000
3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    3.00000
10.00000   10.00000   10.00000   10.00000   10.00000   10.00000   10.00000   10.00000
10.00000    9.00000    8.00000    9.00000    9.00000    5.00000    8.00000   10.00000
12.00000    9.00000    7.00000    7.00000    7.00000    7.00000    6.00000    7.00000
26.00000   23.00000   23.00000   16.00000   12.00000   12.00000   12.00000   22.00000
3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    3.00000    3.00000
10.00000   13.00000    4.00000   10.00000    3.00000    7.00000   11.00000    8.00000
15.00000   14.00000    9.00000   14.00000   15.00000   15.00000    7.00000    7.00000
2.00000    3.00000    4.00000    4.00000    4.00000    2.00000    3.00000    2.00000
2.00000    1.00000    3.00000    2.00000    2.00000    3.00000    1.00000    2.00000
3.00000    4.00000    2.00000    4.00000    2.00000    5.00000    5.00000    4.00000
19.00000    6.00000   10.00000   13.00000   15.00000   13.00000   11.00000   15.00000
0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000    0.00000
7.00000    8.00000   12.00000    9.00000    4.00000    9.00000    7.00000    4.00000

1. Is there any difference between two cohorts due to the intervention?
2. Given the small sample size, will the difference (if any found) be statistically significant?
3. Can we conclude anything meaningful from this data? (before and after the intervention?

I have some background in machine learning and time-series analysis, so I can pick concepts, but I lack experience working with longitudinal data and to do sophisticated statistical analysis. I'm reading the book of Peter Diggle, on analysis of longitudinal data, so I hope to get a grasp of the material.

I would very much appreciate if you can publish the code (R, Python, Matlab) how to analyse this dataset, so I can learn from this explicit example.

• Time series analysis enables one to empirically identify both the number of possible interventions and the ARIMA model appropriate for the data. One should not fit an ARIMA moel and then identify the intervention as the ARIMA model identification phase premises no interventions yielding an inefficient/flawed model. Sophisticated analysis requires more comprehensive approaches. Why don't you post 2 or 3 representative series and I will report some results. There is an R program available to do be used called AUTOBOX ( which I had helped develop) which might be of some use to you. – IrishStat Sep 22 '16 at 18:24
• Can you please send me an email with an excel file attached or post a column oriented version of the data. – IrishStat Sep 29 '16 at 19:58
• @IrishStat, please see the amended example. Does it help or do you want me to send you the excel file? Thanks. – Arnold Klein Sep 29 '16 at 20:54
• pls send excel file ..... – IrishStat Sep 29 '16 at 21:13

You would think that a simple question like this would have received more attention in the literature but .... To determine an anomaly one needs to have a model which characterizes typical behavior. I took your 336 values and simply graphed them and obtained very little visual support for any activity on or around period 169 BUT simple visual checking is equivalent to a simple mean model. . I then used AUTOBOX (my tool of choice which I had helped develop) to simultaneously to identify an appropriate memory model and any exceptional activity. Following is a graph of the actual,fit and forecast using that model. and model suggesting three level shifts and some 14 anomalies while incorporating a very significant AR(1) component, Any thorough analysis of time series data includes a plot of the residuals presented here and the acf of the residuals reflecting/suggesting apparent model sufficiency. In conclusion it appears to me (and AUTOBOX ) that period 169 is not suggestive of any exceptional activity. Hope this helps you and other interested readers.

• wow, that's absolutely amazing! I am wondering, if you have a built-in algorithm which helps you detect a period of "exceptional activity"? – Arnold Klein Oct 21 '16 at 6:20
• Yes we do .. It detects a number of kinds of periods of "exceptional activity" viz 1) pulses .. one time anomalies ; 2 ) Level/step shifts ; 3) Seasonal Pulses ; 4) Local time trends ; 5) changes in parameters and 6) changes in error variance. – IrishStat Oct 21 '16 at 12:36

Step 1: Structure your data in long format:

  ID | time | grp | value |
s_1| 1    |  0  | Q_11  |
s_1| 2    |  0  | Q_12  |
s_1| 3    |  0  | Q_13  |
s_1| 4    |  1  | Q_14  |
s_1| 5    |  1  | Q_15  |
s_1| 6    |  1  | Q_16  |


Step 2: Fit some mixed effects models (linear, logistic, etc. depending on the distribution of your outcome). Let's assume a linear mixed effects regression in R:

library("nlme")

*Below will tell you if the means are different before versus after intervention.
lme(value~factor(grp), data=Your_Dataset, random= ~1|ID)

*Below will tell you if a linear slope of time is different before versus after intervention.
lme(value~factor(grp) + time + factor(grp)*time, data=Your_Dataset, random= ~time|ID)
*Specifically the interaction of grpXtime tells you if the slopes differ.


More examples here: http://dornsife.usc.edu/psyc/20150423gc3/

• How does your suggested approach deal with 1 time anomalies (pulses) in the data. ? How does it deal with the memory model (ARIMA) that may exist in the residuals from a model ? How does it deal with non-constant error variance over time ? How does it deal with unknown points of intervention ? – IrishStat Sep 22 '16 at 21:34
• "How does your suggested approach deal with 1 time anomalies (pulses) in the data?" It doesn't. Heteroscedastic error and auto-regressive properties can be added to the residual structure of the mixed model to handle your other concerns. If by "unknown points of intervention" you mean non-discrete time at intervention, this can also be modeled as a fixed/random effect instead of grp. Your points are well taken, perhaps the OP can clue us into the substantive question to determine if a standard longitudinal model or time-series model is most appropriate. – Nick Jackson Sep 23 '16 at 19:30
• What I meant is that how does the proposed proceedure (automatically ) detect level shift changes and/or time trend changes in the data – IrishStat Sep 23 '16 at 20:03
• Please see the updated topic. – Arnold Klein Sep 29 '16 at 17:09