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Consider the figure below. If the true value is the center of the target, the measured responses in the first instance may be considered reliable, precise or as having negligible random error, but all the responses missed the true value by a wide margin. A biased estimate has been obtained. In contrast, the target on the right has more random error in the measurements, however, the results are valid, lacking systematic error. The average response is exactly in the center of the target. The middle target depicts our goal: observations that are both reliable (small random error) and valid (without systematic error).

Accuracy for a Sample Size of 5

precision vs accuracy

Bias, confounding and effect modification in epidemiology

When examining the relationship between an explanatory factor and an outcome, we are interested in identifying  factors that may modify the factor's effect on the outcome (effect modifiers). We must also be aware of potential bias or confounding in a study because these can cause a reported association (or lack thereof) to be misleading. Bias and confounding are related to the measurement and study design.  Let 's define these terms:

  • Bias: A systematic error in the design, recruitment, data collection or analysis that results in a mistaken estimation of the true effect of the exposure and the outcome.
  • Confounding: A situation in which the effect or association between an exposure and outcome is distorted by the presence of another variable. Positive confounding (when the observed association is biased away from the null) and negative confounding (when the observed association is biased toward the null) both occur.

  • Effect modification : a variable that differentially (positively and negatively) modifies the observed effect of a risk factor on disease status.  Different groups have different risk estimates when effect modification is present..

If  the method used to select subjects or collect data results in an incorrect association, .

THINK >> Bias! 

If an observed association is not correct because a different (lurking) variable is associated with both the potential risk factor and the outcome, but it is not a causal factor itself, 
THINK >> Confounding! 

If an effect is real  but the magnitude of the effect is different for different groups of individuals (e.g., males vs females or blacks vs whites).

THINK >> Effect modification!

Bias Resulting from Study Design

Bias limits validity (the ability to measure the truth within the study design) and generalizability (the ability to confidently apply the results to a larger population) of study results. Bias is rarely eliminated during analysis. There are two major types of bias:

1. Selection bias: systematic error in the selection or retention of participants

Examples of selection bias in case-control studies:

  • Suppose you are selecting cases of rotator cuff tears (a shoulder injury). Many older people have experienced this injury to some degree, but have never been treated for it. Persons who are treated by a physician are far more likely to be diagnosed (and identified as cases) than persons who are not treated by a physician. If a study only recruits cases among patients receiving medical care, there will be selection bias.
  • Some investigators may identify cases predicated upon previous exposure. Suppose a new outbreak is related to a particular exposure, for example, a particular pain reliever. If a press release encourages people taking this pain reliever to report to a clinic to be checked to determine if they are a case and these people then become the cases for the study, a bias has been created in sample selection. Only those taking the medication were assessed for the problem. Ascertaining a case based upon previous exposure creates a bias that cannot be removed once the sample is selected.
  • Exposure may affect the selection of controls – e.g, hospitalized patients are more likely to have been smokers than the general population. If controls are selected among hospitalized patients, the relationship between an outcome and smoking may be underestimated because of the increased prevalence of smoking in the control population.
  • In a cohort study, people who share a similar characteristic may be lost to follow-up.  For example, people who are mobile are more likely to change their residence and be lost to follow-up.  If length of residence is related to the exposure, than our sample is biased toward subjects with less exposure.
  • In a cross-sectional study, the sample may have been non-representative of the general population. This leads to bias. For example, suppose the study population includes multiple racial groups but members of one race participate less frequently in the type of study. A bias results.

2. Information bias (misclassification bias): Systematic error due to inaccurate measurement or classification of disease, exposure or other variables.

  • Instrumentation - an inaccurately calibrated instrument creating systematic error
  • Misdiagnosis - if a diagnostic test is consistently inaccurate, then information bias would occur
  • Recall bias - if individuals can't remember exposures accurately, then information bias would occur
  • Missing data - if certain individuals consistently have missing data, then information bias would occur
  • Socially desirable response - if study participants consistently give the anwer that the investigator wants to hear, then information bias would occur

Misclassification can be differential or non-differential.

Differential misclassification: the probability of misclassification varies for the different study groups, i.e., misclassification is conditional upon exposure or disease status.

Are we more likely to misclassify cases than controls? For example, if you interview cases in-person for a long period of time, extracting exact information while the controls are interviewed over the phone for a shorter period of time using standard questions, this can lead to a differential misclassification of exposure status between controls amd cases. 

Nondifferential misclassification: the probability of misclassification does not vary for the different study groups; is not conditional upon exposure or disease status, but appears random. Using the above example, if  half the subjects (cases and controls) were randomly selected to be interviewed by the phone and the other half were interviewed in person, the misclassification would be nondifferential.

Either type of misclassification can produce misleading results.


Confounding and Confounders

Confounding: A situation in which a measure of association or relationship between exposure and outcome is distorted by the presence of another variable. Positive confounding (when the observed association is biased away from the null) and negative confounding (when the observed association is biased toward the null) both occur.

Confounder: an extraneous variable that wholly or partially accounts for the observed effect of a risk factor on disease status.. The presence of a confounder can lead to inaccurate results.

A confounder meets all three conditions listed below:

  1. confoundingIt is a risk factor for the disease, independent of the putative risk factor.
  2. It is associated with putative risk factor.
  3. It is not in the causal pathway between exposure and disease.

The first two of these conditions can be tested with data. The third is more biological and conceptual.

Confounding masks the true effect of a risk factor on a disease or outcome due to the presence of another variable. We determine identify potential confounders from our:

  1. Knowledge
  2. Prior experience with data
  3. Three criteria for confounders

Example of Confounding

Hypothesis: Diabetes is a positive risk factor for coronary heart disease

We survey patients as a part of the cross-sectional study asking whether they have coronary heart disease and if they are diabetic. We generate a 2 × 2 table (below):

output

'0' indicates those who do not have coronary heart disease, '1' is for those with coronary heart disease; similarly for diabetes, '0'  is the absence and '1'  the presence of diabetes.

The prevalence of coronary heart disease among people without diabetes is 91 divided by 2340, or 3.9% of all people with diabetes have coronary heart disease. Similarly the prevalence among those with diabetes is 12.04%. Our prevalence ratio, considering whether diabetes is a risk factor for coronary heart disease is 12.04 / 3.9 = 3.1. The prevalence of coronary heart disease in people with diabetes is 3.1 times as great as it is in people without diabetes.

We can also use the 2 x 2 table to calculate an odds ratio as shown above:

( 2249 × 26) / ( 91 × 190) = 3.38

The odds of having diabetes among those with coronary heart disease is 3.38 times as high as the odds of having diabetes among those who do not have coronary heart disease.

Which of these do you use? They come up with slightly different estimates.

It depends upon your primary purpose. Is your purpose to compare prevalences? Or, do you wish to address the odds of dibetes as related to coronary health status?

Now, let's add hypertension as a potential confounder.

1) Ask: "Is hypertension a risk factor for CHD (among non-diabetics)?"

First of all, prior knowledge tells us that hypertension is related to many heart related diseases. Prior knowledge is an important first step but let's test this with data.

We consider the 2 × 2 table below:

output

We are evaluating the relationship of CHD to hypertension in non-diabetics. You can calculate the prevalence ratios and odds ratios as suits your purpose. 

These data show that there is a positive relationship between  hypertension and CHD in non-diabetics. (note the small p-values)

2) This leads us to our next question, "Is diabetes (exposure) associated with hypertension?"

We can answer this with our data as well (below):

output

Again, the results are highly significant! Therefore, our first two criteria have been met for hypertension as a confounder in the relationship between diabetes and coronary heart disease.

3) A final question, "Is hypertension an intermediate pathway between diabetes (exposure) and development of CHD?" – or, vice versa, does diabetes cause hypertension which then causes coronary heart disease?  Based on the biology, that is not the case. Diabetes in and of itself can cause coronary heart disease. Using the data and our prior knowledge, we conclude that hypertension is a major confounder in the diabetes-CHD relationship.

What do we do now that we know that hypertension is a confounder?

Stratify....let's consider some stratified assessments...

Stratification and Adjustment - Diabetes and CHD relationship confounded by hypertension: A cross-sectional study - Example

Earlier we arrived at a crude odds ratio of 3.38.

output

Now we will use an extended Maentel Hanzel method to adjust for hypertension and produce an adjusted odds ratio When we do so, the adjusted OR = 2.84.

The Mantel-Haenszel method takes into account the effect of the strata, presence or absence of hypertension.

If we limit the analysis to normotensives we get an odds ratio of 2.4.

output

Among hypertensives we get an odds ratio of 3.04.

output

Both estimates of the odds ratio are lower than the odds ratio based on the entire sample. If you stratify a sample, without losing any data, wouldn't you expect to find the crude odds ratio to be a weighted average of the stratified odds ratios?

This is an example of confounding - the stratified results are both on the same side of the crude odds ratio.This is positive confounding because the unstratified estimate is biased away from the null hypothesis. The null is 1.0. The true odds ratio, accounting for the effect of hypertension, is 2.8 from the Maentel Hanzel test.  The crude odds ratio of 3.38 was biased away from the null of 1.0. (In some studies you are looking for a positive association; in others, a negative association, a protective effect; either way, differing from the null of 1.0)

This is one way to demonstrate the presence of confounding. You may have a priori knowledge of confounded effects, or you may examine the data and determine whether confounding exists. Either way, when confounding is present, as  in this example, the adjusted odds ratio should be reported. In this example, we report the odds-ratio for the association of diabetes with CHD = 2.84, adjusted for hypertension.

If you are analyzing data using multivariable logistic regression, a rule of thumb is if the odds ratio changes by 10% or more,  include the potential confounder in the multi-variable model. The question is not so much the statistical significance, but the amount the confounding variable changes the effect.  If a variable changes the effect by 10% or more, then we consider it a confounder and leave it in the model.

We will talk more about this later, but briefly here are some methods to control for a confounding variable (known a priori):

  • randomize individuals to different groups (use an experimental approach)
  • restrict / filter for certain groups
  • match in case-control studies
  • analysis (stratify, adjust)

Controlling potential confounding starts with good study design including anticipating potential confounders.

Effect Modification (interaction)

Effect modification: occurs when the effect of a factor is different for different groups.  We see evidence of this when the crude estimate of the association (odds ratio, rate ratio, risk ratio) is very close to a weighted average of group-specific estimates of the association.  Effect modification is similar to statistical interaction, but in epidemiology, effect modification is related to the biology of disease, not just a data observation.

In the previous example we saw both stratum-specific estimates of the odds ratio went to one side of the crude odds ratio. With effect modification, we expect the crude odds ratio to be between the estimates of the odds ratio for the stratum-specific estimates.

Effect modifier: variable that differentially (positively and negatively) modifies the observed effect of a risk factor on disease status.

Consider the following examples:

1) The immunization status of an individual modifies the effect of exposure to a pathogen and specific types of infectious diseases. Why?

2) Breast Cancer occurs in both men and women. Breast cancer occurs in men at approximately a rate of 1.5/100,000 men. Breast cancer occurs in women at approximately a rate of 122.1/100,000 women. This is about an 800 fold difference. We can build a statistical model that shows that gender interacts with other risk factors for breast cancer, but why is this the case? Obviously, there are many biological reasons why this interaction should be present. This is the part that we want to look at from an epidemiological perspective. Consider whether the biology supports a statistical interaction that you might observe.

Think about it!  Think about it!

Why study effect modification? Why do we care?

  • to define high-risk subgroups for preventive actions,
  • to increase precision of effect estimation by taking into account groups that may be affected differently,
  • to increase the ability to compare across studies that have different proportions of effect-modifying groups, and
  • to aid in developing a causal hypotheses for the disease

If you do not identify and handle properly an effect modifier, you will get an incorrect crude estimate.The (incorrect) crude estimator (e.g., RR, OR) is a weighted average of the (correct) stratum-specific estimators. If you do not sort out the stratum-specific results, you miss an opportunity to understand the biologic or psychosocial nature of the relationship between risk factor and outcome.

To consider effect modification in the design and conduct of a study:

  1. Collect information on potential effect modifiers. 
  2. Power the study to test potential effect modifiers - if a priori you think that the effect may differ depending on the stratum, power the study to detect a difference.
  3. Don't match on a potentially important effect modifier - if you do, you can't examine its effect.

To consider effect modification in the analysis of data:

  1. Again, consider what potential effect modifiers might be.
  2. Stratify the data by potential effect modifiers and calculate stratum-specific estimates of the effect of the risk on the outcome; determine if effect modification is present. If so,
  3. Present stratum-specific estimates. Use Breslow-Day Test for Homogeneity of the odds ratios, from Extended Mantel-Haenszel method, or -2 log likelihood test from logistic regression to test the statistical significance of potential effect modifiers and to calculate the estimators of exposure-disease association according to the levels of significant effect modifiers. Alternatively, if assumptions are met,  use proportional hazards regression to produce an adjusted hazards ratio.

Example: Diabetes as a Risk for Coronary Heart Disease

When you combine men and women the crude odds ratio = 4.30.

outputoutput

Stratifying by gender, we can calculate different measures. Look at the odds ratios above. The odds ratio for women is 6.66, compared to the crude odds ratio of 4.30. Therefore, women are at much greater risk of diabetes leading to the incident coronary heart disease. For men, the odds ratio is 2.23.

Is diabetes a risk for incident heart disease in men and in women? Yes. Is it the same level of risk? No. For men the OR is 2.23, for women it is 6.66. The overall estimate is closer to a weighted average of the two stratum specific estimates. Gender modifies the effect of diabetes on incident heart disease. We can see that numerically because the crude odds ratio is more representative of a weighted average of the two groups.

What is the most informative estimate of the risk of diabetes for heart disease? 4.30 is not very informative of the true relationship. What is much more informative is to present the stratum specific analysis.

During data analysis, major confounders and effect modifiers can be identified by comparing stratified results to overall results.

In summary, the process is as follows: 
  1. Estimate a crude (unadjusted) estimate between exposure and disease.
  2. Stratify the analysis by any potential major confounders to produce stratum-specific estimates.
  3. Compare the crude estimator with stratum-specific estimates and examine the kind of relationships exhibited.
  • With a Confounder:
    • the crude estimator (e.g. RR, OR) is outside the range of the two stratum-specific estimators ( in the hypertension example - the crude odds ratio was higher than both of the stratum specific ratios);
    • If the adjusted estimator is importantly (not necessarily statistically) different (often 10%) from the crude estimator, the “adjusted variable” is a confounder. In other words, if including the potential confounder changes the estimate of the risk by 10% or more, we consider it important and leave it in the model.
    • Statistical methods (Extended Mantel-Haenszel method, multiple regression, multiple logistic regression, proportional hazards) are available to calculate the “adjusted” estimator, accounting for confounders.
  • With Effect modifiers:
    • the crude estimator (e.g. RR, OR) is closer to a weighted average of the stratum-specific estimators;
    • the two stratum-specific estimators differ from each other
    • Report separate stratified models or report an interaction term.

To review, confounders mask a true effect and effect modifiers means that there is a different effect for different groups.

You have reached the end of the reading material for Week 3!!!  Go to the Week 3 activities in ANGEL.