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Seasonal Interaction Plots

    Monthly Airline Passengers - Series G

  1. Open Monthly Airline Passengers - Series G.xlsx. Click the Month Year for Interaction Plot tab. This is the Series G data from Box and Jenkins, monthly total international airline passengers for January 1949 to December 1960. Month and Year columns have been added and calculated using the Excel date functions =MONTH() and =YEAR().

  2. Click SigmaXL > Statistical Tools > Two-Way ANOVA. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.

  3. Select Monthly Airline Passengers, click Numeric Time Series Data (Y) >>. Select Month for Group Category Factor (X1) >> and Year for Group Category Factor (X2) >>. Uncheck all options: Remove Interaction (Fit Additive Model), Display Residual Plots, Display ANOM Normal Two-Way Chart and Adjust chart alpha for family-wise error rate. Use the default Confidence Level = 95%.

    Seasonal Interaction 3
  4. Click OK. We will not use the ANOVA report. Scroll down to the Interaction Plots. Resize to view the full legend, double click on each Y axis and set Minimum to 0.

    Seasonal Interaction 4a

    Seasonal Interaction 4b
    In the first interaction plot (with Month on the X axis) we can see the monthly seasonal effect and how it gets stronger by year. The second interaction plot (with Year on the X axis) shows the same increasing seasonal effect but we can also clearly see the strong positive trend by year.

  5. Now we will create Seasonal Interaction Plots for Ln (Airline Passengers). Click Recall SigmaXL Dialog menu or press F3 to recall last dialog. Select Ln (Airline Passengers) and click Numeric Data Variable (Y) >>.

    Seasonal Interaction 5

  6. Click OK. Scroll down to the Interaction Plots. Resize to view the full legend, double click on each Y axis and set Minimum to 4 and Maximum to 7.

    Seasonal Interaction 6a

    Seasonal Interaction 6b

    In these interaction plots with Ln (Airline Passengers), we can see that the variability due to monthly seasonal effect is consistent and the yearly trend is more linear. Bisgaard and Kulahch point out that using a traditional interpretation of interaction plots, the similar slopes indicate that the Ln transformation has effectively removed the month by year interaction, so the month and year effect is now additive.

    Bisgaard and Kulahchi (2011) give a novel use of two-way interaction plots to view trends and seasonal effects in data. We will use Two-Way ANOVA to reproduce the interaction charts given in the book.

    Note, in order to produce these charts, the data must be balanced, e.g., every year must have 12 months of data.

    Reference: Bisgaard, S. and Kulahchi, M. (2011), Time Series Analysis and Forecasting by Example, Wiley, pp.111-115.

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