A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data

The overall goal of this statistical trigonometric regression is to model the timing of relapse events that commonly characterize a patient’s disease course in multiple sclerosis and use these models to study seasonal and latitudinal correlates of relapse onset. This method can help explore key questions in the natural history in the epidemiology of multiple sclerosis, including the independent influences of season and latitude, and the timing of relapse, which is great potential to guide subsequent investigations into the biological mechanisms of relapse. The main advantage of this technique is that it provides a flexible regression based tool for exploring and investigating periodic and cyclic phenomena.

This allows the influences of latitude and season to be isolated from a series of other correlates of relapse, including patient and treatment factors. The analysis described here are made using command line driven status software. Begin by opening a do file.

Click on the new do file editor button. Next, using the generate command, calculate the number of relapse onsets dated to each of the 12 calendar months. Do this for the geographic level being modeled, which in this case is the hemisphere.

Execute the generate command as all commands are executed by clicking the execute do do file action button in the due file. Next, execute an S WIL command or an SK test command to test the underlying distribution of relapse counts for normality. These options apply either a Shapiro Wilk or a modified Harkey Bara test respectively.

Make the appropriate choice if the relapse count data is significantly skewed. Then apply a log transformation. If the data passes.

The normality test proceed by using the generate command to create a new variable northco month for Southern Hemisphere. Calendar months offset by six units. Now use the north month variable to make a scatter plot of all the data.

Put the calendar, months and seasons on the x-axis and the observed monthly relapse onsets with relapse frequency on the Y axis. Observe the patterns in relapse onset over the calendar year by viewing each plot in the graph viewer. Next, execute the radar command to draw radar plots of the distribution of relapse frequency by calendar month.

With each radar axis capturing a single month ordered in a clockwise manner. To complete this analysis, execute the CT command to apply an Edwards test of seasonality across the observed relapse data. Repeat this test on each geographic level proceeding with the software.

Build a model of the data. First, specify the annual cycle sign and cosign trigonometric functions to be used in the data regression. Next, use the regress command to specify the form of the base model with the relapse count as the dependent outcome.

This command inputs the just calculated sign and co-sign terms as the primary explanatory variables. Then adds the location specific UVR as an additional adjusting covariate, and it applies the analytic weight option to weight the model for the number of patients contributed by each location. Now, store the model predicted monthly log relapse using the predict command for each site.

Convert log relapses back to integer relapse counts. Exponentiate the log relapse term using the generate command for each site. Then overlay the prediction of the expon monthly relapse estimates using the two-way scatter command.

Now, expand the model by adding an additional harmonic sign co-sign pair using the regress command. Do this two more times, so three sign cosign pairs are added. Each pair will be used to produce a standalone model of the data to estimate the peak relapse probability.

Start with using nl com to calculate the point, estimate and 95%confidence interval for the phase shift. For the best fitting model, convert the outputted point estimates and associated confidence intervals to data numbered for each day of the year. Report the peak relapse frequency or tmax and trough relapse frequency or T in.

Then match the tmax and T in data to a calendar date via the Excel lookup file. This section outlines how to model the ultraviolet radiation data or UVR data. To begin run the use command to load the UVR data.

Then using the egen command, calculate the median monthly UVR for each location. For for each location, use the two-way scatter function to graph a scatterplot of the monthly UVR on the Y axis against the calendar month on the X axis. Then check the normality of the data using s wilk or SK test as before.

Next, using the regress command, specify a base model of location level annual UVR trend. Specify the monthly UVR as the dependent outcome variables, and then make the explanatory variables the previously generated sign and cosign trigono metric functions. Then plot the results and overlay the predicted values.

This involves rerunning the two-way scatter command to overlay the predicted and observe data and using the regress command to run the expanded harmonic model alternatives. Now with the model of best fit, use the generate command to calculate the phase shift point estimate and the associated 95%confidence interval for the UVR. By applying the double angle formulae, again, calculate the team in for each location as previously calculated.

Continue the analysis of the seasonal UVR data by using the merge command to add the previously calculated trough and replace the peak dates for each location. Then use the generate command to calculate the time lapsed in months between the UVR trough date and subsequent relapse peak date. Before proceeding, execute a test for normality on the uuv lag data using the SK test command.

Following the merge, convert the relative latitude to absolute latitude using the ABS X function. Continue using regress to specify a linear mean regression model with UVR trough to relapse peak lag as the dependent outcome variable and absolute latitude in units of 10 degrees as the predictor variable. Be sure to wait the model for the number of patients contributed by each location using the A weights regress option.

Lastly, use the two-way scatter command to plot the absolute latitude on the Y axis against the UVR trough to relapse lag on the x axis displayed in months from the graphing options. Overlay a line of best fit using the L fit graph option, and use the a weights option to visualize the relative patient weights of each location. The trigonometric regression analysis was performed using a data set of 32, 762 relapse events.

Basic plotting of the data suggested an annual cycle with a spring peak and autumn trough across all geographical levels. Relapses peaked a global level in the month of May. In Northern hemisphere data, a corresponding peak in November was observed in the Southern hemisphere data.

These results showed a periodic temporal variation at all three levels of geography best described by an annual cycle with a single peak and a single trough on a six month interval. Thus, this was established as the base case model across both hemispheres. Next UVR regression plots were prepared to determine the UVR trough relapse peak.

The UVR trough relapse peak metric showed the influence of latitude on relapse probability when the latitude from the equator increases. There is a correlative decrease in the lag between UVR trough and subsequent relapse peak After its development. This technique has paved a way to allow researchers to mathematically explore epidemiological phenomena, which varies periodically and psychically.

After watching this video, you should now have a good idea of how to study and critique periodic cyclic phenomena using a trigonometric based regression tool. When applying this technique, it is important to ensure that the data set is clean and the code is debugged. It is also important to overlay and apply a clinical and biological context, particularly in terms of interpreting that the results and the data.