Banner for NCI Division of Cancer Epidemiology and Genetics Biostatistics Branch
Age-Period-Cohort Web Tool

Age Period Cohort Analysis

Age-Period-Cohort analysis identifies patterns in cancer incidence or mortality rates from population-based Count (numerator) and Population (denominator) data. Often the data come from a Cancer Registry (e.g., SEER) in the form of a table showing the numbers of cancer cases or cancer deaths (counts) and corresponding person-years at risk (population) for particular age groups and calendar time periods. In cancer research, the Age-Period-Cohort (APC) framework is a fundamental model to analyze these data. The APC Model includes parameters that describe the mathematical relationships between the Rate of cancer and attained age, calendar period (year of diagnosis), and birth cohort (year of birth). The cancer Rate is expressed as the number of cancers per 100,000 persons at risk, which is calculated from the data using the formula: (Count/Population) x 100,000.

This web tool uses the inputted count and population data to fit the APC Model and returns a number of Outputs. To use this tool, the width of the age and period intervals must all be equal. When this is so, the diagonals of the rate table represent birth cohorts.

Visit the following sections to learn more about the APC web tool:

Getting started

Input data for the web tool consist of Count and Population data for particular age groups over calendar time, in the form of a matrix of rows with paired columns. Rows correspond to particular age groups and columns correspond to calendar time periods. The age and period intervals must all be equal, i.e. if 5-year age groups are used then 5-year calendar periods must also be used. The data can be input by copy-and-paste from an Excel worksheet or file upload of a comma-separated-values (csv) file.

To input from Excel:

  1. Copy the paired columns of data you want to analyze from your spreadsheet, right-click inside the empty matrix on the Input tab, and paste your selection.
  2. Fill in the information (meta-data) on the left hand side of the Input page:
    • Title - describe your data
    • Description - add optional details
    • Start Year - list the first calendar year of the first calendar period of your data, for example, use 1990 for the interval 1990 - 1994
    • Start Age - list the first age of the first age group of your data, for example use 30 for the interval 30 - 34
    • Interval (Years) - the width of the age and period intervals, for example use 1 for single-year data, 2 for two-year data, 5 for five-year data (e.g., 1990 - 1994), etc.
  3. Click the Calculate button.

To input data from a csv file:

The csv file can include only count and population data, or it can include information fields in header lines delimited by semicolons, followed by the count and population data. Click here to see an example.

  1. Click the Browse button and select your file.
  2. Add or modify the meta-data on the left hand side of the input page.
  3. Click the Calculate button.

Changing the default reference age, period, and cohort

Default references

By default, the webtool uses the median age and period ranges as reference points for calculations. The reference range is calculated by the following formulas:

  • 1Reference Age = (Number of Age Groups + 1)/2
  • 2Reference Period = (Number of Periods + 1)/2
  • 3Reference Cohort = (Reference Period – Reference Age + Number of Age Groups)

Let’s use the Sample Data 1 as an example:

  • Reference Age = (7+1)/2 = 4, so the 4th age range is the reference age range
  • Reference Period = (7+1)/2=4, so the 4th period group is the reference period range
  • Reference Cohort = (4-4+7)=7, so the 7th cohort group is the reference cohort range

Since these are all ranges, the reference points are calculated by the following formula:

  • 4Period and Age Points: (Lower Value + Upper Value +1)/2
  • 5Cohort Point: [(Lower Period Value + Upper Period Value +1)/2] – [(Lower Age Value +Upper Age Value +1)/2], or more simply
  • 6Cohort Point: (Reference Period Point – Reference Age Point)

Using the Sample Data 1 as an example:

  • Age: 4th age range is 65-69, so (65+69+1)/2 = 67.5
  • Period: 4th period range is 1950-54, so (1950+1954+1)/2 = 1952.5
  • Cohort: 1952.5-67.5 = 1885

Changing the default:

To change this default to your own selection, after loading your data, change the radio button under References from Automatic to Manual. The drop-down menu’s for Age and Period will contain all possible reference points for the loaded data. Simply choose which of them you want as the reference points for Age and Period, this will calculate the reference point for Cohort from formula 6 above.

Sample data

Illustrative Examples

You can demo the web tool by cutting and pasting the example data into the web tool, then clicking the Calculate button.

Sample Data 1
Title Prostate Cancer Mortality in Nonwhites Copy all data below:

177	301000	271	317000	312	353000	382	395000	321	426000	305	473000	308	498000
262	212000	350	248000	552	279000	620	301000	714	358000	649	411000	738	443000
360	159000	479	194000	644	222000	949	222000	932	258000	1292	304000	1327	341000
409	132000	544	144000	812	169000	1150	210000	1668	230000	1958	264000	2153	297000
328	76000	509	94000	763	110000	1097	125000	1593	149000	2039	180000	2433	197000
222	37000	359	47000	584	59000	845	71000	1192	91000	1638	108000	2068	118000
108	19000	178	22000	285	32000	475	39000	742	44000	992	56000	1374	66000
                                
Description
Example from: Holford T.R. The estimation of age, period, and cohort effects for vital rates. Biometrics, 1983; 39:311-324.
Start Year 1935
Start Age 50
Interval 5
The csv file is found here: Prostate Cancer Example File
Sample Data 2
Title Belgium Female Lung Cancer Mortality Copy all data below:

3	1578947.368	2	1538461.538	7	1400000.000	3	1578947.368	10	1428571.429
11	1666666.667	16	1632653.061	11	1527777.778	10	1408450.704	7	1228070.175
11	1410256.41	22	1666666.667	24	1632653.061	25	1524390.244	15	1136363.636
36	1348314.607	44	1392405.063	42	1660079.051	53	1568047.337	48	1221374.046
77	1590909.091	74	1321428.571	68	1379310.345	99	1636363.636	88	1288433.382
106	1606060.606	131	1541176.471	99	1294117.647	142	1340887.63	134	1285988.484
157	1515444.015	184	1533333.333	189	1490536.278	180	1255230.126	177	986072.4234
193	1307588.076	232	1417226.634	262	1455555.556	249	1414772.727	239	999581.765
219	1066731.612	267	1181415.929	323	1297188.755	325	1335799.425	343	1048929.664
223	849847.561	250	902527.0758	308	1010830.325	412	1115322.144	358	930595.269
198	591574.5444	214	636715.2633	253	688060.9192	338	773632.4102	312	690265.4867
                                
Description
Example from: Clayton D. & Schifflers E. Models for temporal variation in cancer rates. I: Age-period and age-cohort models. Stat. Med., 1987; 6:449-467.
Start Year 1955
Start Age 25
Interval 5
The csv file is found here: Lung Cancer Example File
Sample Data 3
Title US White Female Breast Cancer Mortality Copy all data below:

45	5415162.455	37	5891719.745	27	6293706.294	19	6690140.845	28	6796116.505	21	7000000	20	7194244.604	22	7213114.754	20	7092198.582	15	6849315.068
66	4958677.686	64	5531547.105	78	5967865.34	57	6390134.529	63	6508264.463	55	6740196.078	59	7065868.263	62	7217694.994	43	7251264.755	62	7085714.286
103	4575744.114	138	5126300.149	143	5583756.345	145	6021594.684	114	6263736.264	124	6595744.681	128	6885422.27	123	7089337.176	138	7236497.116	113	7220447.284
172	4269049.392	201	4713883.677	206	5174579.251	195	5609896.433	226	6052490.627	205	6507936.508	220	6644518.272	229	6852184.321	226	7080200.501	222	7264397.906
256	4055124.347	254	4362761.937	301	4768694.55	317	5164548.713	314	5729927.007	410	6220603.854	358	6348643.376	346	6591731.758	346	6881463.803	336	7151979.566
334	3936822.254	341	4110910.187	359	4394124.847	410	4711019.189	427	5243123.772	412	5630722.974	549	6003937.008	562	6349565.021	499	6698885.756	493	6873954.267
442	3894273.128	433	3932788.374	455	4080717.489	460	4303086.997	536	4764444.444	606	5029045.643	612	5630174.793	756	6062550.12	732	6449339.207	761	6526586.621
661	3993957.704	631	3929016.189	587	3942243.116	567	4061604.585	572	4400000	713	4605943.152	729	5192307.692	856	5638998.682	992	6048780.488	869	6242816.092
905	4178208.68	792	4040816.327	750	3928758.512	786	3939849.624	782	4124472.574	824	4309623.431	961	4738658.777	995	5147439.214	997	5560513.107	1225	5993150.685
1205	4339214.98	1123	4168522.643	1013	3986619.441	878	3914400.357	1005	3958251.28	950	4077253.219	985	4356479.434	1144	4703947.368	1184	5083726.921	1351	5638564.274
1501	4442142.646	1408	4287454.324	1328	4122943.185	1251	4009615.385	1139	3948006.932	1167	3937246.964	1147	4081850.534	1237	4340350.877	1348	4638678.596	1417	5141509.434
1945	4509622.073	1826	4396821.575	1707	4274981.217	1851	4647250.816	1463	3795071.336	1291	3874549.82	1367	3884626.314	1412	4041213.509	1484	4243637.403	1701	4657721.796
2118	4453322.119	2260	4443570.586	2192	4394546.913	2038	4281512.605	1875	4120879.121	1779	3963020.717	1523	3869410.569	1632	3891273.247	1728	3986159.17	1728	4295302.013
2348	4309838.473	2529	4439178.515	2464	4482444.97	2389	4411004.431	2256	4271109.428	2122	4139680.062	1981	3973921.765	1913	3848320.257	1839	3832847.02	1977	4024017.912
2634	4157853.197	2654	4358679.586	2695	4461920.53	2823	4448471.478	2657	4370784.669	2545	4287398.922	2421	4094368.341	2207	3888301.621	2185	3790112.749	2195	3846827.9
2825	4012214.174	2808	4168027.312	2807	4264661.197	3019	4330177.854	3117	4371055.953	3026	4366522.367	2710	4208074.534	2739	4026756.836	2432	3894939.142	2299	3802514.059
2827	3849400.871	2994	3946223.804	2977	4028416.779	3111	4167448.091	3211	4325744.308	3271	4404200.889	3221	4303848.21	3047	4173972.603	2810	4032721.01	2682	3817793.594
3000	3687768.9	3006	3774011.299	3015	3855498.721	3258	4000982.439	3170	4175997.892	3407	4291472.478	3478	4280615.385	3423	4229058.562	3159	4109535.58	3147	3903013.767
2744	3524727.039	3025	3623188.406	3087	3709890.638	3175	3817941.318	3183	3937894.346	3431	4070953.963	3626	4169253.766	3662	4215979.738	3676	4139173.516	3496	4023941.068
2614	3101198.244	3065	3450799.37	3058	3549210.771	3351	3645164.799	3301	3748154.877	3421	3872537.922	3637	4028131.576	3916	4134290.541	3869	4117271.47	3750	4086748.038
2770	3120423.567	2996	3260774.924	2962	3382436.908	3216	3516291.275	3473	3623747.913	3462	3737853.595	3623	3855075.548	3866	3944093.042	3992	4008837.116	3906	4041804.636
2649	2894765.599	2905	3058538.64	3059	3203476.804	3231	3370188.797	3237	3900939.986	3546	3581456.419	3584	3650809.82	3685	3713594.679	3977	3852935.478	4008	3946047.061
2382	2689398.216	2737	2827187.274	2797	2969844.978	3181	3138938.228	3177	3260802.628	3562	3361963.19	3642	3437794.978	3912	3505376.344	3888	3638745.905	4097	3740869.248
2374	2493173.703	2599	2573012.573	2662	2699523.375	2995	2854283.808	3078	2991835.148	3282	3102079.395	3511	3212847.731	3766	3299745.904	3935	3375943.72	4016	3452544.704
2310	2277881.866	2614	2328523.071	2598	2437605.555	2722	2569621.448	2975	2713673.265	3206	2830405.226	3344	2962962.963	3600	3060964.204	3746	3111812.593	4021	3177400.237
2279	2037368.139	2366	2106293.955	2502	2196663.74	2513	2292046.698	2777	2427447.552	3000	2548203.517	3258	2679276.316	3495	2790419.162	3627	2871733.967	3919	2955282.407
2169	1787686.475	2290	1880594.564	2277	1956521.739	2428	2017784.426	2448	2137617.883	2710	2256452.956	2938	2382033.404	3388	2514099.139	3346	2612633.716	3670	2714095.548
1946	1538826.506	2023	1634879.586	2237	1705290.441	2208	1747388.414	2256	1850090.208	2403	1957318.563	2740	2078119.075	2868	2194506.083	3155	2306117.974	3294	2407542.757
1723	1286300.859	1834	1375121.841	1884	1453367.276	2047	1494596.963	1998	1574592.166	2219	1657702.077	2340	1767238.124	2610	1872981.701	2863	1963918.233	3175	2056746.777
1424	1018816.627	1539	1117484.752	1711	1220312.389	1848	1278185.088	1920	1322314.05	1915	1366393.15	2167	1444859.315	2308	1527768.584	2419	1614173.228	2732	1711350.539
                                
Description
Tarone RE and Chu KC, Evaluation of birth cohort patterns in population diseases rates. Am J Epidemiol 1996; 143: 85-91.
Start Year 1970
Start Age 24
Interval 2
The csv file is found here: Breast Cancer Example File

What it does

The web tool applies the APC Model to the data in order to estimate parameters (trends and deviations). The parameters are combined to produce functions that describe relationships between the observed Rate of cancer and attained age, calendar period, and birth cohort. The web tool also calculates a number of statistical hypothesis tests (Wald Tests), which address whether the Rate of cancer is statistically significantly variable according to age, period, and cohort factors.

The most important functions calculated by the web tool are summarized in the Table of Key Functions using the following conventions:

  • The APC model is defined over A age groups and P calendar periods with equal intervals.
  • The central age group, calendar period, and birth cohort define standard reference values a0, p0 and c0, respectively.
  • When there is an even number of age, period, or cohort categories, the reference value is the lower of the two central values.
  • All values labeled CI Lo are lower 95% confidence limits. All values labeled CI Hi are upper 95% confidence limits.

Table of Key Functions

Nomenclature Interpretation
Fitted Temporal Trend Expected rates over time in reference age group a0 adjusted for cohort effects
Net Drift Annual percentage change of the expected age-adjusted rates over time
Local Drifts Annual percentage change of the expected age-specific rates over time
Cross-Sectional Age Curve (Cross Age) Expected age-specific rates in reference period p0 adjusted for cohort effects
Longitudinal Age Curve (Long Age) Expected age-specific rates in reference cohort c0 adjusted for period effects
Period Rate Ratios (PeriodRR) Ratio of age-specific rates in each period relative to reference period p0
Cohort Rate Ratios (CohortRR) Ratio of age-specific rates in each cohort relative to reference cohort c0

Statistical hypothesis tests calculated by the web tool are summarized in the Table of Hypothesis Tests.

The Wald Tests follow a Chi-Square distribution when the Null Hypothesis is true. The df (degrees of freedom) count the number of free parameters included in each test. The web tool reports P-values; values less than 0.05 are often considered "statistically significant", meaning there is statistical evidence that the Null Hypothesis is unlikely to be correct.

Table of Hypothesis Tests

Null Hypothesis Implications Degrees of Freedom
Net drift = 0 Fitted temporal trends are stable (i.e., flat with no change) over time.
Fitted longitudinal and cross-sectional age curves are proportional.
1
All age deviations = 0 Fitted longitudinal and cross-sectional age curves are log-linear (i.e., log-additive). A - 2
All period deviations = 0 Fitted temporal trends and period rate ratios are log-linear (i.e., log-additive). P - 2
All cohort deviations = 0 Cohort rate ratios are log-linear; all local drifts equal the net drift. C - 2
All period rate ratios = 1 Net drift is 0 and fitted temporal trends are constant;
Cross-sectional age curve describes age incidence pattern in every period.
P - 1
All cohort rate ratios = 1 Net drift is 0 and all local drifts are 0;
Longitudinal age curve describes age incidence pattern in every cohort.
C - 1
All local drifts = the net drift Temporal patterns are the same in every age group. A-1 if A=P, A otherwise

*For APC model defined over A age groups, P calendar periods, and C = P + A - 1 birth cohorts.

Saving your results

You can save a complete record of the inputs and outputs using the Download button:

Frequently Asked Questions: