95% CI for age-adjusted rate, distrate does not work

[Josna Rani]

Hi,


I am calculating the age-standardised cancer incidence rates for multiple states. I have used the distrate command to determine the ASR for each state along with the 95% confidence interval for each. This worked well.
I also want to present the 95% CI for the age-adjusted rates that is the incidence rate in each age group (Crude rate* age-specific population weight). The distrate command does not give a breakdown of the results by age group.

Any help will be appreciated.

Here is a subset of my data:

  State agegroup cases_2007_2020 avgannualcases population_2007_2020 Crude_R Segi_pop Segi_pwt age_sp_rate_Segi A 20-39 2 0.142857 100662 1.98685 28000 0.4666667 0.9271967 A 40-59 12 0.857143 81640 14.6987 21000 0.35 5.144538 A >=60 8 0.571429 31425 25.4574 11000 0.1833333 4.667197 A All age 22 1.57143 213727 10.2935 60000 1 10.73893 B 20-39 7 0.5 96525 7.25201 28000 0.4666667 3.384271 B 40-59 19 1.35714 113303 16.7692 21000 0.35 5.869216 B >=60 8 0.571429 28711 27.8639 11000 0.1833333 5.108378 B All age 34 2.42857 238539 14.2534 60000 1 14.36187 C 20-39 32 2.28571 238552 13.4143 28000 0.4666667 6.259993 C 40-59 58 4.14286 144346 40.1812 21000 0.35 14.06343 C >=60 19 1.35714 48404 39.2529 11000 0.1833333 7.196374 C All age 109 7.78571 431302 25.2723 60000 1 27.5198

Here is the distrate command:

 
 distrate cases_2007_2020 population_2007_2020 using SEGi_pop_3groups.dta, standstrata(agegroup) popstand(Popwt)///   mult(100000) format(%8.1f) formatn(7)

Here is the result:

State	cases_2007_2020	N	UScrude	USrateadj	USlb_gam	USub_gam	USse_gam	USlb_dob	USub_dob	USsrr	USlb_srr	USub_srr
A	22	213727	10.3	10.7	6.7	16.3	2.4	6.7	16.3	0.5	0.3	0.8
B	34	238539	14.3	14.4	9.7	20.4	2.6	9.7	20.4	0.7	0.5	1
C	109	431302	25.3	27.5	22.5	33.4	2.7	22.5	33.4	1.3	1.1	1.6

(Reposting this for some help with it; the previous post got no response)

[Andrea Discacciati]

Conditional on age group (ie, within each age group), there is nothing to age-standardize.
You could present state- and age-specific rates (ie, rates conditional on both variables. No standardization here). From there, the state-standardized rates (as in the table above) would be obtained as weighted averages of the age-specific rates (ie, you would marginalise over a given age distribution).

[Josna Rani]

Thanks Andrea. Yes, nothing to standardize. That's why I asked about the age-adjusted rates. How to calculate their 95% CI?

The Fay and Fuer method uses gamma distribution. I tried to implement this formula in state but that did not give valid results.

[Andrea Discacciati]

The age-adjusted rates (or age-standardised rates) are those presented in the 5th column of your table.
The CI are reported in column 6-7 and 9-10, using 2 different methods (see -help distrate-)

[Josna Rani]

Thanks Andrea. I think my question was not clear.
I know the distrate gave the overall age-stardised rates. What I need, in addition to that, are the age-adjusted rates. For example, For state A, the ASR is 10.7 (95% CI 6.7-16.3). How do I find the 95% CI for state A age groups 1, 2, and 3? Basically, they are the intermediate steps when we multiply the crude rate by population weight (also called the age-adjusted rate), which is then summed to give the age-standardized rate.

[Andrea Discacciati]

I was probably unclear in #2.
At any rate, to do what you want, you can simply condition (stratify) on state and age group.
This would answer your question

How do I find the 95% CI for state A age groups 1, 2, and 3?

Example:
Download the ancillary files for the distrate package ( net describe distrate, from(http://fmwww.bc.edu/RePEc/bocode/d) )
and then, based on the example in the help file ( help distrate )

use SuffolkCounty.dta, clear
collapse (sum) deaths pop,by(cindpov agegr)
list
distrate deaths pop using year2000st, stand(agegr) by(cindpov agegr) mult(100000)

Directly standardized rates (per 100000)
CI based on the gamma distribution (Fay and Feuer, 1997. Tiwari and al., 2006)

  +------------------------------------------------------------------------------------------------------------------------------------+
  |  cindpov   agegr   deaths        N       crude     rateadj      lb_gam      ub_gam      se_gam         srr      lb_srr      ub_srr |
  |------------------------------------------------------------------------------------------------------------------------------------|
  |   0-4.9%    0-14        1    10608   9.4268477   9.4268477   .23866712   52.523033   13.331576           1           .           . |
  |   0-4.9%   15-24        5     9984   50.080128   50.080128   16.260881   116.87031   24.534152      5.3125   .59445993   251.26817 |
  |   0-4.9%   25-44       54    29190   184.99486   184.99486   138.97383   241.37817   25.406641   19.624255   3.3757414   789.28898 |
  |   0-4.9%   45-64      106    16710   634.35069   634.35069   519.35437   767.22888   61.903533   67.291921   11.819512   2682.6459 |
  |   0-4.9%     65+      657    15825   4151.6588   4151.6588   3840.2271   4481.6201   162.09485   440.40796   78.769933   17421.327 |
  |------------------------------------------------------------------------------------------------------------------------------------|
[ ... output omitted ...]

(The crude and adjusted rates are the same, since conditional on age, there is nothing to age-standardise for. The next step to obtain age-standardised rates, would be, as you said, to calculate a weighted average of those age-specific rates by state (or cindpov, in my example)).

[Josna Rani]

Thanks Andrea. I also tried to put the age-group in the by (var), and the results were the same as crude rates. That is why I was confused.
Shouldn't it be the sum of crude rate * population weight for that age group? And therefore, a different 95% CI?
I am using two different standard populations SEGI and US 2000. They have different population weights. I want to show the difference in age-adjusted rates and the age-standardised rates. Using the distrate command, both std populations would give the same age-adjusted rates because it is showing the crude rate. That seems not right.

[Andrea Discacciati]

The formula for the Age-standardised Rates (ASR) is

ASR = Sum_j CR_j * w_j

where CR_j is the Crude Rate for the j-th age-strata and w_j is the standardisation weight for the corresponding j-th strata.

My initial confusion was that I have never seen the *individual components* of the formula above (ie, the CR_j * w_j) referenced to as "age-adjusted rates". In fact, to me, they are *not* age-adjusted rates by themselves, and most importantly each of them lacks any *meaningful* interpretation (and so do their CIs). They are just intermediate pieces to get the ASRs. Do you have any reference that defines the CR_j*w_j "age-adjusted rates"? Terminology can be confusing, but to me, "age-adjusted rates" is generally used as a synonym for "age-standardised rates". But I am curious to know if in other fields these terms are used differently.

You say you are using 2 different sets of population weights. This will give you different results for the Age-Standardised Rates for the 3 States because the w_j's are different, clearly. But what would the CIs for CR_j*w_j tell you?

The official Stata -dstdize- command has a richer output. It will give you the CR_j*w_j (s*P), but no CIs.

. webuse hb
. generate pop = 1
. dstdize hbp pop age, by(city)
(5 observations excluded because of missing values)

----------------------------------------------------------
-> city= 1
                         -----Unadjusted-----  Std.
                                Pop.  Stratum  Pop.  
  Stratum       Pop.     Cases  Dist. Rate[s] Dst[P]  s*P
----------------------------------------------------------
  15 - 19         91         3  0.257 0.0330  0.220 0.0072
  20 - 24        103         2  0.291 0.0194  0.362 0.0070
  25 - 29         95         4  0.268 0.0421  0.237 0.0100
  30 - 34         65         4  0.184 0.0615  0.181 0.0112
----------------------------------------------------------
Totals:          354        13    Adjusted Cases:     12.5
                                      Crude Rate:   0.0367
                                   Adjusted Rate:   0.0354
                       95% Conf. Interval: [0.0164, 0.0544]

[... output omitted ...]

Summary of Study Populations:
     city             N      Crude     Adj_Rate       Confidence Interval
 --------------------------------------------------------------------------
        1           354   0.036723     0.035415    [  0.016391,    0.054439]
        2           340   0.023529     0.023044    [  0.007178,    0.038911]
        3           215   0.139535     0.124565    [  0.083669,    0.165462]
        5           216   0.018519     0.025030    [  0.000353,    0.049707]

[Josna Rani]

Hi Andrea,

You are right. These are just age-specific rates multiplied by the population weight. Age-adjusted and age-standardised are the same—apologies for creating the confusion.
In my analysis, because of the differing cancer rates by age group, I want to show the result by age group. However, I am having difficulty producing the 95% CI using the inverted gamma distribution or poison distribution.

[Andrea Discacciati]

If you assume that the number of events (d_i) is Poisson, since the weights (s_i) and the total person-time/population (P_i) is not random, an exact 95% CI for s_i*d_i/P_i (see (2) in #3) is, using the distrate example data:

. cii means `=16710/0.2147/100000' 106, poisson

                                                         -- Poisson  Exact --
    Variable |   Exposure        Mean    Std. Err.       [95% Conf. Interval]
-------------+---------------------------------------------------------------
             |   .7782953    136.1951    13.22844        111.5054     164.724

for cindpov=0-4.9% and agegr=45-64 (where 0.2167 is the standard population weight for that age group and 100,000 is a multiplier for the rate).

That said, the interpretation of this quantity and the usefulness of the CI is still unclear to me.