In order to take the first statistics course, read the course content below, then click on "Take The Test", follow the directions, take the test, have it graded, and print your certificate upon successful completion. Please feel free to contact the author, he is hyperlinked below! Remember, as an OJNI subscriber, you do not have to pay for this course, however, this offer is only valid until the next edition of OJNI is released and Statistics 1.2 will be available.
Author: Arthur Johnson, PhD
Upon completion of this 1 unit course, the learner should be able to:
In the above example statistics tells us that when we expect a 0.5 (50%) proportion of men to women that in 19 of the 20 samples we will find between 18 and 32 women. One out of the 20 samples will be outside these limits. Statistical rule number one is that as long as the observed number is at or within the given limits then the result is due to a variation in the sampling and not to some underlying cause. If the observed number is outside the given limits then we should look for an underlying cause. Keep in mind that an outlying sample may indeed be due to chance and our search for an underlying cause will go unrewarded. This is the risk we accept when using the statistical rule.
Calculations of this sort can easily be used to analyze other proportions and other sample sizes. For example, you may want to compare two drugs used for the treatment of hypertension. Statistical analysis allows us to compare the proportion of hypertensive subjects becoming normotensive with the usual treatment versus the new treatment. The argument also extends to other measures such as sample means and the comparison of means.
Looking for Answers
Here is an example of how we would use statistical analysis to understand the results of an experimental study. In this study of the effects of an accepted and a new anti-hypertensive drug, the investigator assigns medication to the subjects as if they were participating in a clinical trial. The results compare possible improvement with a new medication to improvement with the usual medication. The sample size contains 100 subjects.
The results of the above sampling show no evidence that there is any benefit from the new treatment compared to the old treatment.
The results of the second sampling show clear evidence that the new treatment is superior to the usual treatment. However, results from studies are rarely as clear as the above tables indicate. Consider the following table:
The numbers in Table D indicate that subjects with high cholesterol are more likely to experience cardiac events than persons with low cholesterol. Application of a statistical test allows us to see that the difference cannot be reasonably attributed to chance variation. This observation leads us to an important question - will a decrease in cholesterol lead to a decrease in cardiac events? The question helps us to design an experiment to determine whether reducing cholesterol with diet or drugs will lead to a reduction of cardiac events. It is important to accept that the observational data does not justify or endorse a specific intervention.
Can observational data ever justify or endorse an intervention? The answer is a qualified yes. The relationship of smoking to health is a good example. Setting up a study composed of one group of smokers and one group of non-smokers would not be ethically acceptable. However, a variety of observational studies have been conducted that clearly indicate that smoking is deleterious to health. Using the data collected from these observational studies it is possible to make a good case for intervention.
Looking for Bias
Statistics allows us to create questions from observational data and to find answers from planned experiments. In both instances we seek to understand whether chance or cause is responsible for the behavior of the data. But chance and cause are not the only factors at work in statistical analysis. A third factor called bias can blur the interpretation of the data beyond chance or cause. Bias is defined as unrecognized factors that influence the results so that cause can appear as chance and chance can appear as cause.
A properly conducted experimental program in which the investigator assigns a treatment to a specific group is less likely to be affected by bias. The key component is how the assignment of the treatment is carried out. Statistics provides an assignment procedure that assures that each subject has an equal chance of being assigned to one of the two treatment groups. This type of assignment procedure is called random assignment and is similar to flipping a coin for each subject. If the coin comes up heads, the subject is placed in one group and if the coin comes up tails the subject is placed in a second group. This is the only assignment procedure that protects against bias.
In an observational study, the subjects in effect assign themselves by the traits they possess to a specific group. In this type of study the possibility of bias is always present. This is the main reason why observational studies are used to suggest a question rather than provide an answer.
You may read that more elaborate statistical analysis can adjust (at least for know variables) against any biasing influences. Be assured that this is only possible because such analyses minimize the influence of subjects whose characteristics vary appreciably from the rest of the group by simply ignoring them.
Looking for an Appropriate Number of Subjects
Common sense tells that if we have developed a breakthrough drug we should be able to demonstrate its effectiveness by designing a small study. A less effective drug would require a larger group of subjects to demonstrate the effectiveness of a particular drug. Often the number of subjects in a study will be based on the number of available subjects or on the number of subjects the researchers were able to afford. Studies of this sort are quite likely to have too small a sample size.
Studies with too large a sample size are seldom an issue. However, if a study is too small we might conclude that the results are due to chance when a larger study might show the drug to be effective. Apart from the consequences of failing to find a useful drug the small study raises a critical ethical problem. The participants who give their informed consent to participate in a study are defrauded if the study is too small to yield useful results.
Statistics can be helpful in determining the correct sample size. In order for statistics to be helpful however, the investigator must have a working idea of the magnitude of difference between the new treatment and the usual treatment. If the goal of the study is to determine which drug treatment is better (how much improvement does the new drug provide compared to the usual drug) then how much improvement does the new drug have to provide to be clinically significant.
If the usual drug causes decreased blood
pressure or decreased cholesterol levels in 20% of the subjects, how much more
improvement does the new drug have to provide to justify a change to the new
drug? How much greater a percent resolution would be of clinical importance -
25%, 30%? For example, if the investigator were interested in distinguishing a
20% improvement rate from a 30% improvement rate then the number of subjects per
group would be 400. Once the investigator makes a commitment to the magnitude of
difference, the statistician can easily calculate the sample size for the study.
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