OverviewThis page briefly describes age-period-cohort analysis and provides an annotated resource list. DescriptionAge Period Cohort Effect Age period cohort (APC) analysis plays an important role in understanding time-varying elements in epidemiology. In particular, APC analysis discerns three types of time varying phenomena: Age effects,
period effects and cohort effects. (1) Conventional solutions to APC identification problem Constrained Coefficients GLIM estimator (CGLIM) Guideline for estimating APC models (Based on Yang and Land) (5):
Median Polish Analysis-Practical Example (3) Table (3) demonstrates the identification problem, where the three components (age, period and cohort) are perfectly correlated. To identify the cohorts we need to know only the period and age group: we subtract the early age group from the upper and lower period limit (e.g. people who were 10-14 years old in 1950-1954 we subtract 10 from 1950 and 1954 to label the cohort interval as 1940-1944). (9) The color highlighted diagonal fields indicate the rate for each cohort as they age. Contingency tables cannot estimate mutually exclusive cohort risk because of overlapping cohorts. This convention may introduce misclassification of some individuals, but the primary purpose of an age-period-cohort analysis is to estimate general trends in cohort specific rather than a precise quantification of a “true” causal risk. The overlapping cohort remind us against over interpretation of estimates. We also are limited by missing data. For example we only have one point of data for the youngest group population (those aged 10-14 in 2000-2004). Using this table, we can perform an initial graphical representation with a line graph in Microsoft Excel. The two graphs were created using the line charts in Microsoft Excel. To plot both graphs we just rearranged the data with the “Switch Row/Column” function. These two graphical representation allow us to assessed for any pattern in the data. The limitation is that any finding could represent a mix of two or more effects. Median polish removes the additive effects of age and period by iteratively subtracting the median value of each row and column. (6) The first step in the median polish is to calculate medians for each row, see table 2: The next step is to subtract the row median from each value in the row, for example in row one we will subtract 0.610 minus 0.790 = -0.18. In the second row (15-19 years old) we used the same procedure 6.330 – 5.770 = 0.56,and then for each cell in the table. This created a table with new values, see table 3: The next step is to calculate the column median for the new values, and then subtract the column median from each cell in the column, for example -0.18 – 19.08 = -19.26. After creating the new table with the values from subtracting each median column for each cell, we proceed to calculate the row median (third iteration). These iterations will eventually produce row and column medians equal to zero. For this example, 6 iterations were necessary to produce row and column medians equal to zero, see table 4: Table 4 has the residuals values after 5 iterations.These residuals represent the coefficients free of the additive effect of age and period effects. Notice the data for age groups 75-79 and 80-84 years old between 1910 and 1939 are missing values. If we replace the missing values for zero rates, the calculate residuals will be biased. The full procedure was performed in Microsoft Excel. To check whether these residuals were correct, we created a new table with the product of subtracting the residual value from the original the set of values in table 1. The product of the subtractions are use to create a line chart. This line chart allow us to check the validity of the residuals and we expect lines perfectly parallel. Since we are subtracting the residuals that represent cohort effects from the original values we are assessing for any age or period effect free of cohort effects. See graphs 3 and 4: The median polish procedure is available in R, which is a free available software (8). See the next syntax: mpdata <- read.csv(“C:/Users/mydocs/suicidemp.csv”, header=FALSE, stringsAsFactors=FALSE) Median polish results can be obtain without any transformation of rates, but using the log transformation of the rates before the median polish procedure will produce an assessment of interaction on the multiplicative scale (or log-additive effect). We repeated our median polish procedure using log transformation of the suicide rates. To produce log transformed residuals of the original table using R software we created a new function replacing the rates for log transformed rates (noticed the bold font in the syntax): medpolish2 <- function (x, eps = 0.01, maxiter = 10L, trace.iter = TRUE, na.rm = FALSE) med.p2 <- medpolish2(mpdata, na.rm = TRUE) The data is saved as a comma delimited file (.csv), an easy format to read in R. Notice the command for the median polish, the option of missing data is enable, otherwise the procedure will report an error. Both
sets of residuals created with Excel and R are equal. We calculated the mean for each cohort and then these log transformed residuals are use to create a plot by cohort. This plot helps to assess the distribution of the residuals, where any significant deviation from zero will suggest a strong cohort effect for that cohort, see next graph: STATA code for plotting of the residuals: Plot of rate median polish residuals, BOOK EXAMPLE (log scale) These residuals helps us to assess the magnitude of the cohort effect using a linear regression of the residuals values by cohort. Here we choose the 1910 – 1914 as the reference cohort. Similar to the graph 6, it seems that the cohorts born after 1950 were at statistical significant higher risk for suicide compared to the cohort of 1910-1014. The coefficients calculated with the linear regression are in log scale, to estimate the rate ratios we used the exponent function for each coefficient [exp(x)]. STATA code for the regression of suicide rate residuals. char cohort[omit] 17
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Websiteshttp://yangclaireyang.web.unc.edu/research/age-period-cohort-analysis-new-models-methods-and-empirical-applications/ |