# How can we compare multiple proportions from multiple independent populations to evaluate implementation of a treatment?

I want to compare sixteen Case Fatality Rates (deaths per 100 cases) of a particular disease from sixteen different populations across 7 years. Each population received the same treatment but some regions did not implement it properly. As a result, I am trying to show the effectiveness the treatment had in each of the regions to prove a hypothesis that

More deaths in some regions were because of not giving treatment thoroughly.

Can someone suggest me a way to do this?

Update: Below is the data from 2003 to 2010 if that helps:

SI.    2003              2004              2005               2006            2007              2008           2009            2010
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Cases   Deaths    Cases   Deaths    Cases    Deaths    Cases   Deaths  Cases   Deaths    Cases  Deaths  Cases   Deaths  Cases   Deaths
1      31      4         7       3         34       0         11      0       22      0         6      0       14      0       132     5
2      109     49        235     64        145      52        392     119     424     133       319    99      462     92      562     125
3      6       2         85      28        192      64        21      3       336     164       203    45      325     95      50      7
4      12      5         17      0         6        0         1       0       0       0         0      0       0       0       x       x
5      0       0         0       0         4        0         0       0       27      0         39     0       66      3       80      0
6      104     67        37      27        46       39        2       1       32      18        13     3       12      10      2       1
7      226     10        181     6         122      10        73      3       32      1         3      0       246     8       138     1
S      17      2         9       1         1        0         3       3       2       0         2      0       3       0       19      5
9      475     115       22      0         51       0         1       0       0       0         24     0       1       0       34      17
10     1       0         0       0         1        0         0       0       65      0         4      0       6       0       118     15
11     0       0         0       0         0        0         0       0       7       0         0      0       9       2       11      6
12     0       0         0       0         1        0         0       0       0       0         0      0       0       0       x       x
13     0       0         0       0         0        0         0       0       0       0         12     0       0       0       7       0
14     163     36        88      9         51       11        18      1       37      0         144    0       265     8       458     7
15     1124    237       1030    228       6061     1500      2320    528     3024    645       3012   537     3073    556     3540    494
16     2       1         3       1         12       6         0       0       16      2         58     0       0       0       1       0

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Just to be clear, you've got 4 tables, each one if of 2 by 2, with number of "Fatality" for each population (from the people who got and didn't get treatment). Is that correct, or are you having it in any way different then what I described? –  Tal Galili Jan 28 '11 at 8:16

If I'm reading you right (and changing Tal's 4 to a 5), then at http://en.wikipedia.org/wiki/Statistical_hypothesis_testing if you scroll halfway down you'll find the formula for "Two-proportion z-test, pooled for d0 = 0." I would think you'd want to do such a test for each of the five years, then choose a meta-analytic method of pooling the results.

(You can also use an online calculator for each test. http://www.dimensionresearch.com/resources/calculators/ztest.html and http://www.surveystar.com/our_services/ztest.htm are not perfect but each looks serviceable.)

In light of further comments...From the research question you've posed, it sounds as if regional differences per se are not important. Therefore you could simplify a great deal by collapsing across thorough-treatment regions and not-thorough-treatment regions, yielding two sets of regions for which to test the difference in proportions. You could do this for each of the years on which you have a substantial amount of data. Then you could pool the different years' test results using a standard meta-analytic method, and you would have a single answer to your question of whether the two levels of implementation show significantly different results.

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Thank you. I got the answer. I request you for one more advise. Now I have some more statistics. I want to compare number of deaths and number of cases of a disease from three different populations Each population received a different treatment. I have year wise data from 2003 to 2010. I want to compare the effect of these two treatments for each year and also total eight years and know whether the difference is significant. Kindly advise which is the best statistical test. –  DrWho Jan 29 '11 at 23:27
Interesting. And more complicated. So now your outcome variable has 3 values: healthy, ill, or died? It seems as if you could run chi-square tests separately within each year and compare the residuals or the measures of association across years. Another level of sophistication would involve, for each year, a multinomial logistic regression, and finding a way to compare the coefficients across years. Someone out there may be able to suggest another way using something like loglinear modeling. –  rolando2 Jan 30 '11 at 2:09
Thank you. There is some communication gap. I will clarify. I am getting more data from 3 to 4 regions. Now I have survived and died cases from different regions. One region reported more deaths because of not giving treatment thoroughly. one region reported zero deaths. some regions had only 1 case and no deaths. I have data from 2003 to 2010 year wise and region wise. I am a medical doctor with limited knowledge of statistics. –  DrWho Jan 30 '11 at 4:40
It's difficult to visualize the data that you have. I suggest you post a table with the number who had each outcome, organized by region and by year (not necessary to label the regions of course). That way one of us can give you the most informed recommendation. –  rolando2 Jan 30 '11 at 5:10
@rolando2: sorry about that. I have edited my original post to include a table. Thank you for your time –  DrWho Jan 30 '11 at 5:21