Calculating age- adjusted rates is pretty simple, it just becomes overwhelming to wrap your mind around because there are a stack of age categories, each with a count, two separate populations, and then your standard population. However, breaking it into logical steps can make it less confounding.
you can see a real-life example here as well: Age Adjusted Rate Explaination
Something to keep in mind: you will need to adjust the rate to reflect your desired representative population. This simply means that you multiply your final rate by the representative size, 10,000 or 100,000 or even 1,000 if you are looking at something common which can bear that level of granularity.
This can be done at two separate places in the computation: 1. Right after the crude rate for the age group is made 2. At the very end of the computation after standardization
Which you choose does not really matter globally, it is a personal choice. The reference I have attached does it at the crude rate level. My personal preference is at the very end, and that is simply because you are calculating on a single number at this point, less possible errors, and it makes chasing down problems easier to wrap your mind around should you find there is an inconsistency somewhere.
Also, I am rounding here to make the tables simple and interpretable, but you should not round until the very last step because you will introduce compounded rounding errors reducing the accuracy and precision of your estimate significantly.
You need to be sure that you have all of your counts and populations in the same age groupings. If they are not, you may need to re-aggregate.
If the groups are currently in 5-year groups to 85+ and your summary has 65+, you can simply add the the counts for the 5 groups between 65 - 85+ to create this grouping.
If you are looking at 10-year groups, just combine the first two, and the next two, until you hit your top group.
If, however you will be looking at atypical groupings, such as 2-year, then you will need single year counts and as well as populations for both the local and standardizing demographics.
Note be very careful here. You can add counts, populations and Age distributed weights, but NOT CRUDE RATES. Crude rates are NOT PROPORTIONAL so, when aggregating age groups, do so by aggregating counts and propulations then calculating new rates
I created a sample set of age groups and simulated counts data as well as simulated standardized-population data, so you can see both the equations and the logic surrounding each step of the adjustment process.
The key to this first step is just making sure that the counts and populations are properly aggregated to the same age groupings.
This is pretty straight forward, you are simply dividing the counts for each age group by the population for that same age group.
\[ Count/Population = Crude Rate\]
If you are going to include the estimate size in the crude rate, then do that here. I prefer to wait until the final step, but neither is more correct.
This step is the one that seems to most confuse people, it is really not very difficult if you are thoughtful and logical about it. There are three parts to this.
\[AgeGroup/ \sum { AgeGroup } = Proportion\]
These proportions are how we will weight each local age group of our local population so that the effect you are estimating so that it reflects the age distributions in your standard population.
\[CrudeRate \times Proportion = AgeSpecificAdjustedRate\] \[ AgeSpecificAdjustedRate\_0to14 = .02 \times .123 = .00246 \quad (calculate\ for\ each \ row)\] 2. Add the age- specific adjusted rates together to get the age adjusted rate.
\[\sum { AgeSpecificAdjustedRates } = AgeAdjustedRate \]
\[AgeAdjustedRate = \sum.00246 + .000915 + .005796 + .001638 + .002208 = .013017\]
3. If you did not apply a population size to the crude rate, you do it now, to this final rate. If you are looking for a rate per 100,000 people, then n=100000.
\[AgeAdjustedRate \times n = Final Rate Per_n=.013017 \times 100000 = 1302 *\]Thisreal method for the most precise interval is complex, but you can use the extension of the intervals one would use on a standard normal distribution using the margin of error +/- the mean and for this particular calculation the age adjusted rate serves as the center. So you can calculate as follows You can see justification for this method on this publication Confidence Intervals
\[ MarginOfError + AgeAdjustedRate\]\[ MarginOfError - AgeAdjustedRate\]
\[AgeAdjustedRate/\#Events\]
\[ \#Events= sum(25 + 17 + 41 + 34 + 58) = 175\]
\[AgeAdjustedRate= 1302\]
\[ SE = 1302/175 = 7.44\]
\[MarginOfError = SE \times 1.96 = 7.44 \times 1.96 =14.58 = 15 \ (rounded \ which \ is \ a \ personal \ scientific \ decision)\]
I would round this up, you can decide how to handle rounding of rates and intervals. It is somewhat subjective
\[ UpperConfidence = AgeAdjustedRate + MarginOfError = 1302 + 15 = 1317\] \[ LowerConfidence = AgeAdjustedRate - MarginOfError = 1302 + 15 = 1287\]
So your age adjusted rate and interval for these four fictious age groupings, counts and populations based on our ficticous standard is: 1302 ( 1287, 1317)