Inferential Statistics - Part 1

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Inferential statistics compare two or more groups or study relationships between study variables. In other words, these help to test the hypothesis. Usually, two types of data may be tested. For nominal or ordinal data, where you wish to compare frequencies, Chi-Square test or Fisher’s Exact test are used. If the need is to compare quantitative or continuous data, then there are other tests, where one needs to decide upon the test depending on the hypothesis and data. These are detailed in next sections.?

?Overall, when you have a quantitative data and you have to decide upon a test, your first question should be, is my data parametric or normally distributed. Plotting a graph from individual data points using statistics software can check this. Once you have arrived at a conclusion regarding distribution of the data, it is much easier to choose an appropriate test. However, sometimes, despite an adequate sample size, the data is skewed with outliers, and is non-parametric in distribution as discussed earlier. Whatever is the case, the data distribution generally governs the inferential statistics to be used for analysis. Before deploying a test, one must understand some critical aspects of analysis.?

Type 1 and Type 2 Errors

Type 1 (also denoted as α) and Type 2 (also denoted as β) errors. To understand and remember this, let us assume that Type 1 error is bigger error than Type 2 error (being at number 1 is stronger than number 2). Type 1 error is when the convict was innocent but was punished, i.e. when the null hypothesis was true but rejected (false positive). Type 2 error is when the convict was guilty but was not punished (false negative). For obvious reasons, punishing the innocent (Type I error) is a bigger error than not punishing the guilty.?

Power of the Study

The power of a study (sensitivity to assess the true difference) is the statistical strength of the probability (possibility) with which the test will reject the null hypothesis when the null hypothesis is actually false. This means that there is probability of not committing a Type 2 error, or making a false negative decision. The power of a study is dependent on the sample size. So an adequate sample size increases the probability of not committing Type 2 error or false negative results, thereby increasing the sensitivity of the study.

Hypothesis Testing

Hypothesis refers to an idea that we propose, e.g. XX cancer drug is as good as YY cancer drug. In simple words, it is a “case’’ that you file in court to prove that someone is innocent or guilty, and during the case hearing in court (study for a researcher), various evidences (endpoints) are brought up to prove innocence or guilt (and in not a chance based decision). This process of proving the innocence or guilt is similar to what we do in hypothesis testing. Therefore, a statistical hypothesis test is a method of making decisions using data from a study, where the findings are not dependent on chance alone.?

Null hypothesis is basically an assumption, which assumes there is “null” or no difference between the two groups/treatment/intervention like new drug XX (as above example) and the old established drug YY are same. As a researcher you want to reject a null hypothesis (a natural tendency of human brain). This is as simple as where a court always assumes that a person is innocent, and a lawyer wants to prove that the person is guilty. Therefore, in the light of the data from studies or evidences are used to prove a point. Then one may ask why make a hypothesis that we want to reject. This is because it is always easier to negate and reject the ideas rather than accepting them (you see, human brain works naturally like this). Therefore, during a research study, we always have a null hypothesis and an “alternative” hypothesis. As a basic understanding, null hypothesis is the one that we wish to reject and the alternative hypothesis is the one that we wish to accept and prove.

Statistical Significance or Probability

What does it mean by saying "Significant"? Statistically significant result simply means it is unlikely to have occurred by chance. Generally, it is the P-value that we use to express the statistical significance. To understand it better, consider this example. Suppose testing a relation between smoking and breathlessness yields a P-value of 0.07. This means even if smoking and breathlessness were not associated in the population, there was a 7% chance of finding such an association due to random error in the sample. Therefore, the amount of evidence required to accept that an event is unlikely to have arisen by chance is known as the significance level or critical p-value. Precisely, “probability” (P-value) means that the observed relationship (e.g., between variables) or a difference (e.g., between means) in a sample occurred by pure chance ("luck of the draw"), and that in the population from which the sample was drawn, no such relationship or differences exist. The higher the p-value, the lesser is the reliability of the observed relation between variables in the sample, and vice-versa.

The P-values can be set as one tailed and two tailed. One-sided P-value is used when you are sure that the association is in one direction only. E.g. surgery for gall bladder stones, where you are sure that the surgery will result in clinical benefit. However, when you compare a new drug with an established drug, this can be either better or worse than the established drug. Here you must choose a two-sided P-value because you do not know the direction of the effect.?

Confidence Interval

Confidence interval (CI), confidence level and confidence limits ?provide an estimate of the confidence (90%, 95%? or 99 % as set) with which you can say that there are 90%, 95 % or 99% chances that the mean of the true population will fall within those limits. E.g. if the mean in your sample is 50, and the lower and upper limits of the 95% CI? 40 and 60, respectively, then you can say that there is a 95% probability that the true mean of that particular population will be between 40 and 60. For a 95% CI, probability (p-value) is set to 0.05. If the p-level is set to a smaller value, the range of CI will also widen, which will increase the “confidence” of saying that the true mean is between XX and YY because now your range is wider. E.g. for p=0.01, CI will be 99%, for p=0.05, CI will be 95%. The larger is the sample size, more reliable is the mean value obtained. Now let us see how the confidence intervals tell the difference between the two groups in a study is significant or not, even if you do not have the p-values.

Ex. Group 1, CI is 26-30 and Group 2, CI is 27-32. The numbers 27-30 are common in both ranges and hence there is NO significant difference between the two groups.

Ex. Group 1, CI is 26-30 and Group 2, CI is 31-37. There are NO common numbers between the two groups, and hence there IS A significant difference between the two groups.

Another common usage of CI’s is seen while presenting Odd’s Ratios (discussed later in this chapter). If the CI range for Odd’s Ratio includes 1, it means that there is no significant association between risk factor and outcome.

Coming next - Inferential Statistics - Part 2

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Dr. Payal Bhardwaj