[Biostatistics] Statics Relative Contents - Primer

[TomHsiung]

Three kinds of data.

The heights of Martians and Venusians are known as interval data because heights are measured on a scale with constant intervals, in this case, centimetres. For interval data, the absolute difference between two values can always be determined by subtraction.

There are other kinds of data, such gender, state of birth, or whether or not a person has a certain disease, that are not measured on an interval scale. These variables are examples of nominal or categorical data, in which individuals are classified into two or more mutually exclusive and exhaustive categories. In every case, it is possible to categorise each individual into one and only one category. In addition, there is no arithmetic relationship or even ordering between the categories.

Ordinal data fall between interval and nominal data. Like nominal data, ordinal data fill into categories, but there is an inherent ordering (or ranking) of the categories. Level of heath (excellent, very good, good, fair, or poor) is a common example of a variable measured on an ordinal scale. The different values have a natural order, but the differences or "distances" between adjoining values on an ordinal scale are not necessarily the same and may not even be comparable.

[admin]

All statistical methods are built on the assumption that the individuals included in your sample represent a random sample from the underlying (and unobserved) population. In a random sample every member of the population has an equal probability (chance) of being selected for the sample.

There are two ways to create a random sample, including Simple Random Sample and Stratified Random Samples.

In the way of simple random sample, we create a random sample by to obtain a list of every member of the population of interest, number them from 1 to N, then use a computerized random number generator to select the n individuals fro the sample. In the way of stratified random samples, members are first divided into different subgroups based on something like gender, race, or geographic location, etc., then we construct simple random samples within each subgroup (strata). This procedure is used when there are widely varying numbers of people in the different subpopulations so that obtaining adequate sample sizes in the smaller subgroups would require collecting more data than necessary in the larger subpopulations if the sampling was done with a simple random sample.

[CheneyHsiung]

The median is the value that half the members of the population fall below. Since 50% of the population values fall below the median, it is also called the 50th percentile.

Calculation of the median and other percentiles is simple, the value that defines the lower half of the observations, is simply the .5 (n+1) observation. When there are an odd number of observations, the median falls on one of the observations. For example, if there are 27 observations, the .5 (27+1) = 14th observation (listed from smallest to largest) is the median. When there is an even number of observations, the median falls between two observations. For example, if there are 40 observations, the median would be .5 (40+1) = 20.5th observation. Since there is no 20.5th observation, we take the average of 20th and 21st observation.

Other percentile points are defined analogously. For example the 25th percentile point, the point that defines the lowest quarter of the observations, is just .25 (n+1) observation. Again, if the value falls between two observations, take the mean of the two surrounding observations. In general, the pth percentile point is the (p/100)(n+1) observation.

Computing the percentile points of a population is a good way to see how close to a normal distribution it is. Recall that we said that in a population that exhibits a normal distribution of values, about 95% of the population members fall within 2 standard deviations of the mean and about 68% fall within 1 standard deviations of the mean. For a normal distribution, the value of the associated percentile points are:

0.15th percentile, mean - 3 standard deviation

2.5th percentile, mean - 2 standard deviation

16th percentile, mean - 1 standard deviation

25th percentile, mean - 0.67 standard deviation

50th percentile, (median) mean

75th percentile, mean + 0.67 standard deviation

84th percentile, mean + 1 standard deviation

97.5th percentile, mean + 2 standard deviation

99.85th percentile, mean + 3 standard deviation

If the values associated with the percentiles are not too different from what one would expect on the basis of the mean and standard deviation, the normal distribution is a good approximation to the true population and then the mean and standard deviation describe the population adequately.

Why care if or not the normal distribution is a good approximation? Because many of the statistical procedures used to test hypotheses require that the population follow a normal distribution at least approximately for the tests to be reliable.

[Janis.Y.Chen]

OR can also be calculated in another way, whereas

OR=[EER/(1-EER)]/[CER/(1-CER)]

=

{[A/(A+C)]/[1-A/(A+C)]}/{[B/(B+D)]/[1-B/(B+D)]}.

When the incidence of disease, that is A/(A+C) is small, A/(A+C) is very very small compared with 1 and [1-A/(A+C)] ≈ 1. Therefore, OR ≈ RR. The situation for B/(B+D) is same vice versa.

[TomHsiung]

Hazard Ratio:

In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two levels of an explanatory variable. For example, in a drug study, the treated population may die at twice the rate per unit time as the control population. The hazard ratio would be 2, indicating higher hazard of death from the treatment. Or in another study, men receiving the same treatment may suffer a certain complication ten times more frequently per unit time than women, giving a hazard ratio of 10.

[TomHsiung]

Why you can't calculate risk in a case-control study?
For most people, the risk of some particular outcome, being akin to probability, makes more sense and is easier to interpret than the odds for that same outcome. That being so, maybe it would be more helpful to express the stroke/exercise result in the case-control study in Figure 12.8 as a risk rather than as odds. Unfortunately, we can't and here is why.

To calculate the risk that those with a stroke had exercised, you need to know two things: the total number who'd had a stroke and the number of those who had been exposed to the risk (of exercise). You would then divide the latter by the former. In a cohort study on the other hand, you start with healthy individuals and follow them to measure the proportion exposed to the risk factor who subsequently developed the illness. The proportion would be an estimate of the risk in the population.

However, in a case-control study, you select on the basis of whether people have some illness or condition or not. So you have one group composed of individuals who've had a stroke, and one group who have not had stroke, but both groups will contain individuals who were, and who were not, exposed to the risk. Moreover, you can select whatever number of cases and controls you want. You could, for example, halve the number of cases and double the number of controls. This means that the column totals, which you would otherwise need for your risk calculation, are meaningless. The result of this is that the population at risk cannot be estimated using a case-control study and so risks and risk ratios cannot be calculated. However, there is a way round this problem, the odds ratio.

Significance of OR
Generally speaking OR (and Relative Risk see below) values greater than 1 indicates that a disease is more likely to occur in an exposed group as compared to an unexposed group. Conversely, an OR value less than 1 means that a disease event is less likely to occur in an exposed groups compared to unexposed group. An OR that has a 95% CI that overlaps with 1 is indicative of an OR that is not a significant (Box 5).

[TomHsiung]

The Odds Ratio
With a case-control study, you can compare the odds that those with an illness will have been exposed to the risk factor, with the odds that those who do not have the illness will have been exposed. If you divide the former by the latter, you get the odds ratio. In other words, an odds ratio compares the odds of acquiring the illness if exposed to the risk, with the odds of the illness if not exposed to the risk.