Because the variance in a count or proportion tends to be proportional to the count or proportion itself, theory (and much experience) suggest analyzing the square roots of the data.
See for yourself by plotting the proportions and overall counts on square-root axes.
So that each column has a visual impact directly proportional to the count it represents, the column widths (as well as their heights) are also proportional to the square roots of the counts: this makes the areas of the columns directly proportional to the counts. The columns are only lightly drawn because they are of secondary interest in this visualization of proportions, as the title states.
The apparently random variation of points (representing the proportions) around their smooth (shown as the blue line), as well as the approximate symmetry of that variation around the smooth, attest to the appropriateness of the square root scale. They also suggest that a more sophisticated analysis of temporal correlation is unnecessary: you can be confident that the trends you see in this plot are real. They present a subtler picture than suggested in the question: the proportions do increase, but only through the first seven years.
Creating such a combined plot can be done in Excel or Stata, but is difficult, fussy, and time-consuming in either program. This example was produced with the
ggplot2 package in
R (version 3.4.0).
To illustrate the process, here is the full
X <- data.frame(Year=2003:2016,
scale.dup <- 0.5e6 # Proportional to column heights in the plot
ggplot(X, aes(Year, 100 * scale.dup * Young/All)) +
geom_col(aes(Year, All, width=2.25*sqrt(All/scale.dup)),
fill="#ffffe0", alpha=0.75, color="Gray") +
scale_y_continuous(sec.axis=dup_axis(~. / scale.dup, "Young / All (%)"), trans="sqrt") +
ggtitle("Disease Registry Patient Proportions", "2003 - 2016 (square root scales)")