Relative Values

Statistical Terms n Mean:  the average of the data  sensitive to outlying data n Median:  the middle of the data  not sensitive to outlying data n Mode:  most commonly occurring value n Range:  the difference between the largest observation and the smallest n Interquartile range:  the spread of the data  commonly used for skewed data n Standard deviation:  a single number which measures how much the observations vary around the mean n Symmetrical data:  data that follows normal distribution  (mean=median=mode)  report mean & standard deviation & n n Skewed data:  not normally distributed  (mean  median  mode)  report median & IQ Range

Measures of Frequency of Events n Incidence -The number of new events (e.g. death or a particular disease) that occur during a specified period of time in a population at risk for developing the events. n Incidence Rate -A term related to incidence that reports the number of new events that occur over the sum of time individuals in the population were at risk for having the event (e.g. events/person-years). n Prevalence -The number of persons in the population affected by a disease at a specific time divided by the number of persons in the population at the time.

Measures of Association n Relative risk and cohort studies -The relative risk (or risk ratio) is defined as the ratio of the incidence of disease in the exposed group divided by the corresponding incidence of disease in the unexposed group. n Odds ratio and case-control studies -The odds ratio is defined as the odds of exposure in the group with disease divided by the odds of exposure in the control group.

Measures of Association

n Absolute risk -The relative risk and odds ratio provide a measure of risk compared with a standard. n Attributable risk or Risk difference is a measure of absolute risk. It represents the excess risk of disease in those exposed taking into account the background rate of disease. The attributable risk is defined as the difference between the incidence rates in the exposed and non-exposed groups. n Population Attributable Risk is used to describe the excess rate of disease in the total study population of exposed and non-exposed individuals that is attributable to the exposure. n Number needed to treat (NNT) -The number of patients who would need to be treated to prevent one adverse outcome is often used to present the results of randomized trials.

Relative Values As a result of statistical research during processing of the statistical data of disease, mortality rate, lethality, etc. absolute numbers are received, which specify the number of the phenomena. Though absolute numbers have a certain cognitive values, but their use is limited.

Relative Values In order to acquire a level of the phenomenon, for comparison of a parameter in dynamics or with a parameter of other territory it is necessary to calculate relative values (parameters, factors) which represent result of a ratio of statistical numbers between itself. The basic arithmetic action at subtraction of relative values is division.

In medical statistics themselves the following kinds of relative parameters are used: n Extensive; n Intensive; n Relative intensity; n Visualization; n Correlation.

The extensive parameter, or a parameter of distribution, characterizes a parts of the phenomena (structure), that is it shows, what part from the general number of all diseases (died) is made with this or that disease which enters into total.

Using this parameter, it is possible to determine the structure of patients according to age, social status, etc. It is accepted to express this parameter in percentage, but it can be calculated and in parts per thousand case when the part of the given disease is small and at the calculation in percentage it is expressed as decimal fraction, instead of an integer.

The general formula of its calculation is the following: part × 100 total

n The intensive parameter characterizes frequency or distribution. n It shows how frequently the given phenomenon occurs in the given environment. n For example, how frequently there is this or that disease among the population or how frequently people are dying from this or that disease. n To calculate the intensive parameter, it is necessary to know the population or the contingent.

n General formula of the calculation is the following: phenomenon×100 (1000; ; ) environment

General mortality rate number of died during the year × 1000 number of the population

n Parameters of relative intensity represent a numerical ratio of two or several structures of the same elements of a set, which is studied. n They allow determining a degree of conformity (advantage or reduction) of similar attributes and are used as auxiliary reception; in those cases where it isn’t possible to receive direct intensive parameters or if it is necessary to measure a degree of a disproportion in structure of two or several close processes.

n The parameter of correlation characterizes the relation between diverse values. n For example, the parameter of average bed occupancy, nurses, etc. n The techniques of subtraction of the correlation parameter is the same as for intensive parameter, nevertheless the number of an intensive parameter stands in the numerator, is included into denominator, where as in a parameter of visualization of numerator and denominator different.

n The parameter of visualization characterizes the relation of any of comparable values to the initial level accepted for 100. This parameter is used for convenience of comparison, and also in case shows a direction of process (increase, reduction) not showing a level or the numbers of the phenomenon. n It can be used for the characteristic of dynamics of the phenomena, for comparison on separate territories, in different groups of the population, for the construction of graphic.

Consider a box containing chips or cards, each of which is numbered either 0 or 1. We want to take a sample from this box in order to estimate the percentage of the cards that are numbered with a 1. The population in this case is the box of cards, which we will call the population box. The percentage of cards in the box that are numbered with a 1 is the parameter π. SIMULATION

In the Harris study the parameter π is unknown. Here, however, in order to see how samples behave, we will make our model with a known percentage of cards numbered with a 1, say π = 60%. At the same time we will estimate π, pretending that we don’t know its value, by examining 25 cards in the box. SIMULATION

We take a simple random sample with replacement of 25 cards from the box as follows. Mix the box of cards; choose one at random; record it; replace it; and then repeat the procedure until we have recorded the numbers on 25 cards. Although survey samples are not generally drawn with replacement, our simulation simplifies the analysis because the box remains unchanged between draws; so, after examining each card, the chance of drawing a card numbered 1 on the following draw is the same as it was for the previous draw, in this case a 60% chance. SIMULATION

Let’s say that after drawing the 25 cards this way, we obtain the following results, recorded in 5 rows of 5 numbers: SIMULATION

Based on this sample of 25 draws, we want to guess the percentage of 1’s in the box. There are 14 cards numbered 1 in the sample. This gives us a sample percentage of p=14/25=.56=56%. If this is all of the information we have about the population box, and we want to estimate the percentage of 1’s in the box, our best guess would be 56%. Notice that this sample value p = 56% is 4 percentage points below the true population value π = 60%. We say that the random sampling error (or simply random error) is -4%. SIMULATION

An experiment is a procedure which results in a measurement or observation. The Harris poll is an experiment which resulted in the measurement (statistic) of 57%. An experiment whose outcome depends upon chance is called a random experiment. ERROR ANALYSIS

On repetition of such an experiment one will typically obtain a different measurement or observation. So, if the Harris poll were to be repeated, the new statistic would very likely differ slightly from 57%. Each repetition is called an execution or trial of the experiment. ERROR ANALYSIS

Suppose we made three more series of draws, and the results were + 16%, + 0%, and + 12%. The random sampling errors of the four simulations would then average out to: ERROR ANALYSIS

n Note that the cancellation of the positive and negative random errors results in a small average. Actually with more trials, the average of the random sampling errors tends to zero. ERROR ANALYSIS

So in order to measure a “typical size” of a random sampling error, we have to ignore the signs. We could just take the mean of the absolute values (MA) of the random sampling errors. For the four random sampling errors above, the MA turns out to be ERROR ANALYSIS

The MA is difficult to deal with theoretically because the absolute value function is not differentiable at 0. So in statistics, and error analysis in general, the root mean square (RMS) of the random sampling errors is generally used. For the four random sampling errors above, the RMS is ERROR ANALYSIS

The RMS is a more conservative measure of the typical size of the random sampling errors in the sense that MA ≤ RMS. ERROR ANALYSIS

For a given experiment the RMS of all possible random sampling errors is called the standard error (SE). For example, whenever we use a random sample of size n and its percentages p to estimate the population percentage π, we have ERROR ANALYSIS