58. Assessing Seasonality (24 Averages)

For a proper assessment of seasonality, information of a second year of observation is needed, as well as information not only of, e.g., the months of January and July, but also of adjacent months. In order to unequivocally demonstrate seasonality, all of this information included in a single test is provided by autocorrelation.

The above graph gives a simulated seasonal pattern of C-reactive protein levels in a healthy subject. Lagcurves (dotted) are partial copies of the datacurve moved to the left as indicated by the arrows.

Do repeatedly measured outcome value follow a seasonal pattern.

Primary question: do repeatedly measured CRP values in a healthy subject follow a seasonal pattern. If the datacurve values are averaged values with their se (standard error), then x_i will change into (x_i + se), and x_i+1 into (x_i+1 + se). This is no problem, since the se-values will even out in the regression equation, and the overall magnitude of the autocorrelation coefficient will remain unchanged, irrespective of the magnitude of the se. And, so, se-values need not be further taken into account in the autocorrelation of time series with means, unless they are very large. A data file is given below.

The entire data file is in extras.springer.com, and is entitled “chapter58seasonality”. Start by opening the data file in SPSS. We will first try and make a graph of the data.

The above graph shows that the average monthly C-reactive protein levels look inconsistent. A graph of bi-monthly averages is drawn. The data are already in the above data file.

The above bi-monthly graph shows a rather seasonal pattern. Autocorrelation is, subsequently, used to test significant seasonality of these data. SPSS Statistical Software is used.

For analysis the statistical model Autocorrelations in the module Forecasting is required.

The above graph of monthly autocorrelation coefficients with their 95 % confidence intervals is given by SPSS, and it shows that the magnitude of the monthly autocorrelations changes sinusoidally. The significant positive autocorrelations at the month no. 13 (correlation coefficients of 0,42 (SE 0,14, t-value 3,0, p < 0,01)) further supports seasonality, and so does the pattern of partial autocorrelation coefficients (not shown): it gradually falls, and a partial autocorrelation coefficient of zero is observed one month after month 13. The strength of the seasonality is assessed using the magnitude of r² = 0,42² = 0,18. This would mean that the lagcurve predicts the datacurve by only 18 %, and, thus, that 82 % is unexplained. And so, the seasonality may be statistically significant, but it is pretty weak, and a lot of unexplained variability, otherwise called noise, is in these data.

Autocorrelation is able to demonstrate statistically significant seasonality of disease, and it does so even with imperfect data.

More background, theoretical and mathematical information about seasonality assessments is given in Statistics applied to clinical studies 5th edition, Chap. 64, Springer Heidelberg Germany, 2012, from the same authors.