Background
Standardisation is frequently used in medical research to allow for the influence of differences in case mix (such as different age or sex distributions) when comparing populations or sub-groups (such as different regions or hospitals).
The indirect arithmetic method is the most commonly used standardisation method in the literature. It compares the actual number of events in a local area (e.g. Birmingham) with the number expected when factor-specific event rates (e.g. age, sex) in a reference population (e.g. England) are applied to the local population. This method is often used to look at differences in mortality rates by means of standardised mortality ratios (SMRs) [
1,
2]. It has also been used to assess other events such as NHS performance indicators [
3,
4]. However ratios cannot be directly compared to one another with this method only to the Standard (For example SMR = 100). In addition, indirect standardisation cannot be applied if the number of events in the reference population is unknown.
Direct standardisation, another frequently used method, involves applying local age-sex specific rates to the age-sex population estimate of a reference, or standard [
5‐
7]. This approach enables comparisons between local areas, for example, comparing the incidence of cancer in different regions of England, and allows for the differing age and gender structures in different areas of the country [
8]. This technique therefore depends on the availability of age/sex specific rates for a local population.
For relatively rare conditions, there will be considerable instability in local age/sex-specific rates of disease and indirect standardisation is a more robust method if the populations are small or there is uncertainty about the stability of age-specific death rates [
9].
Logistic regression standardisation, an alternative to the arithmetic methods has advantages over these latter approaches when individual level data are available, through for example, a survey.
Logistic regression allows the effect of variables (e.g. age and sex), and interactions between these factors, on outcomes of interest (e.g. presence of disease) to be estimated. Additional demographic data may be of use and also variables, such as age, could be included as continuous variables in the model, thus having a smoothing effect on the estimates.
Using Poisson regression to model rates and adjust for confounders is not uncommon,[
10] however such modelling does not usually apply a standard population to the models identified. Standardisation using logistic regression modelling involves calculating the sum of the predicted probabilities of the outcome of interest for each individual in the local population and establishing the ratio of the observed and expected event rates [
11]. Examples of the use of regression standardisation include describing variation in practice admission rates [
12]; measuring income related quality of life [
13]; measuring inequity in the delivery of healthcare [
14]; and calculating hospital mortality ratios, adjusting for age, sex, diagnosis, admission method and length of stay [
15,
16].
The equivalence of indirect and logistic regression-based standardisation with a saturated model when adjusting for case-mix has been previously demonstrated [
11]. Nevertheless, the arithmetic direct/indirect methods continue to be the more popular and widely utilised methods employed in health service research. The most probable reasons for this may be the lack of survey data and the perception that logistic regression-based standardisation is more difficult than the arithmetic methods. This paper aims to illustrate the application of logistic regression to calculate standardised smoothed prevalence estimates of disease when the direct method may produce biased estimates and the indirect method is not possible.
Illustrative data
The Birmingham elderly thyroid study (BETS), a cross-sectional survey of people aged 65 years and over has been used to illustrate the methods discussed in this paper. BETS aim was to determine the prevalence of subclinical hypothyroidism and hyperthyroidism in the elderly [
17]. Demographic data were collected from participants and included age and sex. Of the 16,125 patients invited to participate in BETS, only 5,881 (36.5%) took part in the survey. Response rates varied by age (43% 65–69 years to 26% 80+ years) and gender (35% male vs. 40% female). Participants had a different age and sex structure to that in the National population and adjustment was necessary to allow inferences about the prevalence of disease in England and Wales to be made. A standardisation approach was chosen to correct for this response bias [
9].
The crude prevalence of subclinical hyperthyroidism and subclinical hypothyroidism were 2.2% (128/5881) and 2.9% (168/5881) respectively. Age-specific subclinical hyperthyroidism rates ranged from 1.7% (16/945) in males aged 65–69 years to 2.3% (9/388) in males aged 80+.
Methods
To calculate rates for subclinical hyperthyroidism standardised by age and gender by the direct method, ages were categorised into four 5-year age bands (65–69, 70–74, 75–79, 80 and over). The formulae used to calculate the standardised rates are given below:
(i) Direct method
The directly standardised rate is obtained by dividing the total expected number of cases in a standard population by the standard population size
(1.1)
where i = 1 to 4 age groups and j = 1, 2 sexes, N
ij is the standard population size in age group i, sex j,
, p
ij is the age-sex specific rate in the study,
is the estimated age-sex specific rate in the study, n
ij is the age-sex specific population in the study.
The standard error of a directly standardised rate is given by:
(1.2)
Where p
ij are all small, as is often the case, p
ij (1-p
ij) can be replaced with p
ij thus [1.2] reduces to
(1.3)
A 95% confidence interval for the standardised rate (using a normal approximation) is then:
standardised rate ± 1.96 (standard error (standardised rate))
(ii) Logistic regression method
When individual data (presence/absence of disease, age and sex) are available, logistic regression allows us to examine the relationship between the probability of disease (p) and potential explanatory variables via the logit transformation of p:
(2.1)
where p is the age-sex specific rate in the study, α, β, γ and βγ are unknown parameters, age (years), sex (1 = male, 0 = female)
The data can be used to provide estimates (maximum likelihood) of these parameters and hence an estimated
The estimated logit is then weighted by the Standard age/sex specific population sizes (N
agesex)
(2.2)
where N
agesex is the population with a specific age and sex
.
The standardised rate is then obtained by back transformation:
(2.3)
The variance of the standardised logit is given by:
(2.4)
and standard error of the standardised logit is thus:
(2.5)
The 95% confidence interval of the standardised logit is: standardised logit ± 1.96 standard error (standardised logit) = (lower, upper)
Back transforming again to obtain the confidence interval for the standardised rate:
(2.6)
This method of calculating the confidence interval for the standardised logit and then back transforming to obtain standardised rates is used since the distribution of the logit is liable to be closer to the Normal distribution since the scale ranges from (-∞ to +∞) as opposed to between (0 and 1). The price for this benefit is that the estimator is a biased estimator of the statistic in equation
1.1. The bias could be estimated by using equation
1.1 where
is obtained by back transforming the logits.
As with any logistic model building process, the linearity assumption for any continuous variables should be confirmed. A method based on quartiles can be used to test this assumption. A categorical variable with 4 levels is created using three cutpoints based on the quartiles of the distribution of the continuous variable (e.g. age). The model can then be refitted with the categorical variable and a plot of the estimated coefficients versus the midpoints of the quartile groups can be examined to determine linearity [
18]. The effectiveness of the model to describe the outcome variable should also be assessed with the Hosmer-Lemeshow goodness of fit test [
18].
Discussion
This study has illustrated the similarity of standardised rates when calculated by direct standardisation and logistic regression and has demonstrated the value of logistic regression in instances where individual level data are available.
Logistic regression is a practical and intuitive approach to standardisation. Most statistical packages contain regression analysis procedures and the methods described in this paper are suitable for implementation in SAS and STATA (SPSS requires an additional step to obtain case-wise estimates of logit and standard error (logit) [
20]).
Direct standardisation requires categorisation of the population and the rates. If adjustment is necessary for several variables (such as age, sex and deprivation) then some categories may have very low or zero rates, thus generating an imprecise estimate of the standardised rate. Once direct standardisation has been implemented, then calculation of rates is generally a routine method (requiring only the input of category specific numbers of cases) and the potential bias caused by small numbers may be missed. Logistic regression standardisation tends to fail to converge to a solution when the number of cases are too small, alerting the researcher to problems with the data.
The main advantage of the logistic regression method is that it allows adjustment by continuous variables in addition to categorical variables and therefore has the potential to lose less information than the direct method which only allows for standardisation by categorical variables. The allowance of continuous variables also has a beneficial smoothing effect on the model. Logistic regression standardisation can also allow for adjustment by non-linear variables and interactions between variables. The structure of the model can be extended to include random effects [
21]. This may be particularly useful when allowing for clustering effects (e.g. hospitals, general practices), thereby incorporating cluster variation in the standard error of the predicted values. The logistic regression method also allows standardisation when there is missing data through the process of imputation whereas the direct method would exclude these observations from the analysis [
22]. In addition this method will identify the amount of variation explained by the variables and will highlight those that have a significant effect on the outcome, giving the analyst the choice to include or exclude variables [
18]. Nevertheless, to avoid the problem of data dredging any potential variables should be decided on prior to analysis being performed [
23].
Another possible benefit of logistic regression standardisation is that the method may identify the absence of significant variables and consequently demonstrate that there is no requirement or benefit from standardisation.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
AR and RH planned the study. AR conducted the analyses and produced the first draft of the manuscript. RH provided guidance on statistical analysis. All authors contributed to the reviewing and editing of the manuscript. All authors read and approved the final manuscript.