A major reason for performing clinical research is to generalize the findings from the set of observations on one group of subjects to others who are similar to those subjects. “Inferential statistics” implies using statistical methods for making inferences (generalizations) from the results obtained from the sample to the population from where the sample was selected. Two important questions are raised in scientific studies: how reliable are the results obtained (estimation) and how probable is it that the differences between observed & expected results on the basis of the hypothesis have been produced by chance alone (test of statistical significance).<sec id="sec001"><title>Important concepts in inferential statistics

A major reason for performing clinical research is to generalize the findings from the set of observations on one group of subjects to others who are similar to those subjects. “Inferential statistics” implies using statistical methods for making inferences (generalizations) from the results obtained from the sample to the population from where the sample was selected. Two important questions are raised in scientific studies: how reliable are the results obtained (estimation) and how probable is it that the differences between observed & expected results on the basis of the hypothesis have been produced by chance alone (test of statistical significance).<sec id="sec001"><title>Important concepts in inferential statisticsProbability

Assume that an experiment can be repeated many times and assume that one or more outcomes can result from each trial. Probability is relative frequency of the number of occurrences of a given outcome to the total number of occurrences of all possible outcomes. If a given outcome is sure to occur, then probability of that outcome is 1 and if the outcome can not occur, its probability is 0. Probability can be of two types: mathematical and statistical. Mathematical probability is an experiment or a trial where the probabilities of occurrences of various outcomes are already established mathematically (for example probability of getting a head when a coin is tossed). On the other hand in case of statistical probability an experiment or a trial is required to find out the probabilities of occurrences of various outcomes (for example probability of development of chronic lung disease in extremely low birth weight newborns or probability that drug A is better than drug B). It's important to distinguish this concept of objective probability from subjective probability which reflects a person's opinion, hunch or best guess. Calculation of probability (Table 1) of a particular outcome depends upon relationship between various possible outcomes. Two or more events are mutually exclusive if the occurrence of one precludes the occurrence of the others. Two different events are independent if the outcome of one event has no effect on the outcome of the second. Table 1

Calculation of Probability

Relationship between outcomes	Calculation	Probability (P) of outcomes (A, B, C, D)

Mutually exclusive events	Addition rule	P(A or B) = P(A) + P(B)
Independent events	Multiplication rule	P(A and B) = P(A) × P(B)
Non-independent events	Modified multiplication rule	Conditional probability P(A and B) = P(A if B present) × P(B) or = P(B if Apresent) × P(A)
Non-mutually exclusive events	Modified addition rule	P(A or B) = P(A) + P(B) − P(A and B)

Probability distribution:

A series of probabilities associated with various occurrences/outcomes/possibilities of events in an experiment/trial/study will generate a probability distribution. Theoretically several types of probability distributions are possible. Basically three types of probability distributions need to be considered: Binomial, Poisson and Normal distribution. Binomial and Poisson distributions are used for discrete variables and Normal distribution for continuous variables. Binomial distribution gives the probability that a specific outcome occurs in a given number of independent trials. Poisson distribution is used to determine the probability of occurrence of an outcome a specified number of times. Most important probability distribution in statistical inference is Normal distribution (Guassian distribution). Normal distribution will generate a Normal (Guassian) curve. Properties of Normal distribution are:

It is bell shaped & symmetrical

The three types of averages: the mean, the median & the mode will be almost equal.

The total area under the normal curve will be equal to “1”.

Fifty percent of the sample values will lie on the left of the perpendicular drawn on the middle and the remaining 50 % will lie on the right of this line.

Mean − 1 SD & mean + 1 SD (standard deviation) will include about 68 % of the sample values.

Mean − 2 SD& Mean + 2 SD will include about 95 % of the sample values.

Mean − 3 SD & mean + 3 SD will include about 99 % of the sample values.

Theoretically the curve touches the horizontal line only at the infinity

(Sample value − Mean)/SD which is called as Standard Normal Deviate/Z-score is distributed with a mean of “0” and a SD of “1”, whatever the variable may be. This is a very important property. Inference theory is based on this property.

Estimation and hypothesis testing

There are two approaches to statistical inference: estimating parameters and testing hypothesis.

Estimation of population parameters

Estimation can be carried out on the basis of sample values from a larger population. Values of mean, proportion, correlation coefficient etc. computed from sample serve as estimates of the population parameters. This estimate is a single value and is called Point estimate. A good point estimate should be unbiased and should have small variability (minimum variance) from one sample to another. As median has greater variability from one sample to another, mean instead of median is the recommended statistic when distribution of observations follows a normal distribution. If the distribution of observations is skewed, median is the better statistic because median has minimum variability in such a situation. Mean/proportions and standard deviation can be used to estimate intervals: a lower limit (LL) and an upper limit (UL) are computed from sample values. It can be said with a certain amount of confidence (for example 95% confidence intervals) that the population value (true value) of the parameter will lie within these limits. These limits are called Confidence limits or Interval estimates. The LL and UL estimates for the population mean are given as: mean − C x SE and mean + C x SE. [C= Confidence coefficient, SE = {SD/(n)}, n = sample size. If 95% confidence is desired, C = 1.96; for 99% confidence, C = 2.58; for 99.9% confidence, C = 3.29].

Hypothesis testing

Hypothesis testing enables one to quantify the degree of uncertainty in sampling variation, which may account for the results that deviate from the hypothesized values in a particular study. It is important to distinguish between a research hypothesis and a statistical hypothesis. The research hypothesis is a general idea about the nature of the clinical question in the population of interest. The primary purpose of the statistical hypothesis is to establish the basis for tests of significance.

A general procedure is that of calculating the probability of observing the difference between two values if they really are not different. This probability is called the P value, and this condition is called the null hypothesis (H₀). On the basis of the P value and whether it is low enough, one can conclude that H₀ is not true and that there really is a difference. In the following text, we elaborate on these key concepts and the definitions needed to understand the process of hypothesis testing. There are few steps necessary for conducting a statistical hypothesis test:

Formulate the null (H₀) and alternative (H₁) hypotheses

Decide on the appropriate test statistic and compute the test statistic for the given condition

Select the level of significance for statistical test

Determine the value the test statistic which must attain to be declared significant

Calculate the resulting P value

Either reject or do not reject H₀ (reject H₀ if the P value is less than or equal to a prespecified significance level [typically 0.05]; do not reject H₀ if the P value is greater than this significance level)

Interpret the results according to the clinical hypothesis relevant to H₀ and H₁. Each of these steps is discussed in the following text.

<sec id="sec007"><title>Null and alternative hypothesis:

In general, H₀ assumes that there is no association between the predictor and outcome variables in the study population. Given the H₀ of no association between a predictor variable and an outcome variable, a statistical hypothesis test can be performed to estimate the probability of an association due to chance that is derived from the available data. Thus, one never accepts H₀, but rather one rejects it with a certain level of significance. In contrast, H₁ makes a claim that there is an association between the predictor and outcome variables. One does not directly test H₁, which is by default accepted when H₀ is rejected on the basis of the statistical significance test results. If the evidence is insufficient to reject the null hypothesis, it is retained but not accepted per se.

One and two sided tests:

The investigator must also decide whether a one-or two-sided test is most suitable for the clinical question. A one-sided H₁ test establishes the direction of the association between the predictor and the outcome—for example, that the prevalence of retinopathy of prematurity (ROP) is higher in preterm babies receiving oxygen therapy. In this example, the predictor is oxygen therapy and the outcome is ROP. However, a two-sided H₁ test establishes only that an association exists without specifying the direction—for example, the prevalence of ROP in premature babies receiving oxygen therapy is different (i.e. either higher or lower) from that in premature babies without oxygen therapy. In general, most hypothesis tests involve two-sided analyses.

Test statistic:

The test statistic is a function of summary statistics computed from the data. A general formula for many such test statistics is as follows:

Where the relevant statistics and standard error are calculated on the basis of the sample data. The standard error is the indicator of variability, and much of the complexity of the hypothesis test involves estimating the standard error correctly. H₀ is rejected if the test statistic exceeds a certain level (i.e. critical value). Test statistic is selected based on probability distribution.

Select the level of significance:

The level of significance when chosen before the statistical test is performed is called the alpha value (α). This gives the probability of incorrectly rejecting the null hypothesis and to keep this possibility small, usually a value of 0.05 (sometimes 0.01 or 0.001) is chosen.

Determine the value the test statistic must attain to be declared significant (Critical value), calculate the resulting P value and either reject or do not reject Ho:

Each test statistic has a distribution. The distribution of the test statistic is divided into an area of acceptance and an area of rejection. The critical value is the dividing line between these areas.

For example, for continuous data, the Student t test is most often used to determine the statistical significance of an observed difference between mean values with unknown variances. On the basis of large samples with underlying normal distributions and known variances, the Z test of two population means is often conducted. Similar to the t test, the Z test involves the use of a numerator to compare the difference between the sample means of the two study groups with the difference that would be expected with H₀, that is, zero difference. The denominator includes the sample size, as well as the variances, of each study group. Once the Z value is calculated, it can be converted into a probability statistic by means of locating the P value in a standard reference table. The Figure-1 illustrates a standard normal distribution (mean of 0, variance of 1) of a test statistic, Z, with two rejection regions that are either below −1.96 or above 1.96. Two hypothetical test statistic values, −0.5 and 2.5, which lie outside and inside the rejection regions, respectively, are also included. Consequently, one does not reject H₀ when Z equals −0.5, but one does reject H₀ when Z equals 2.5. Figure 1Graph illustrates the normal distribution of the test statistic Z in a two-sided hypothesis test. Under H₀, Z has a standard normal distribution, with a mean of 0 and a variance of 1. The critical values are fixed at ±1.96, which corresponds to a 5% significance level (ie, type Ierror) under H₀, The rejection regions are the areas marked with oblique lines under the two tails of the curve, and they correspond to any test statistic lying either below −1.96 or above +1.96. Two hypothetical test statistic values, −0.5 and 2.5, result in not rejecting or rejecting H₀, respectively.

Interpret the results according to the clinical hypothesis relevant to H0 and H1:

Interpret the results and state the conclusion in words.

Example

Arya V B et al (1) conducted a randomized controlled trial to answer the research question that “whether oral phenobarbitone (intervention) causes a 50% (effect size) reduction in incidence of neonatal hyperbilirubinemia (outcome) in term ‘high risk’ newborns (population)”. Note that this is a one-sided hypothesis, which specifies the direction of association between intervention and outcome variable i.e. reduction in neonatal hyperbilirubinemia with use of phenobarbitone. We will analyze the data from this study assuming it to be two sided hypothesis i.e. “whether oral phenobarbitone (intervention) causes a 50% (effect size) alteration in incidence of neonatal hyperbilirubinemia (outcome) in term ‘high risk’ newborns (population)” The study had 37 babies in phenobarbitone group, out of which 6 babies developed hyperbilirubinemia (Total Serum Bilirubin > 13 mg%) and mean value of TSB at 72±12 hrs was 10.0±3.7 mg/dl. In placebo group (n=38) 13 babies developed hyperbilirubinemia and mean value of TSB at 72±12 hrs was 12.3±3.3 mg/dl. Table-2 describes comparison of two proportions. Table 2

Comparing two independent proportions by using Z test


Step 1	State H₀ and H₁	H₀: No difference in proportion of hyperbilirubinemia in phenobarbitone and placebo groupsH₁: 50% difference in proportion of hyperbilirubinemia in phenobarbitone and placebo groups
Step 2	One or two sided test	Two sided test
Step 3	Test used	z test
Step 4	Select the level of significance	α = 0.05
Step 5	Determine critical value	For two sided α = 0.05, looking at reference table [available in standard textbooks of biostatistics (2), the value of z distribution that separates the upper and lower 2.5% (α = 0.05 divided into 2] from central 95% will be ±1.96. We will therefore reject the null hypothesis if the calculated z statistic is less than critical value of −1.96 or greater than +1.96.
Step 6^@	Calculate test statistic	z = p₁-p₂/Standard error of difference of proportions = p₁-p₂/square root{p(1-p)[(1/n₁)+(1/n₂)]} Calculations according to data given above will result in z statistic of −1.82
Step 7	Accept or reject null hypothesis	As calculated z statistic falls in the central 95% (Figure 1) we can not reject the null hypothesis
Step 8	State the conclusion	Phenobarbitone does not reduce the incidence of hyperbilirubinemia in the investigated sampling frame.

^@p1 and p2 are proportions in two groups with sample size of n1 and n2. p is pooled proportion

The result can also be reported in terms of confidence interval. For above mentioned example confidence interval can be calculated as follows

For 95% CI value of Confidence coefficient is 1.96

Thus for above said data

As this 95% confidence limits cross 1, true difference can be anywhere from −0.37 to +0.01. Therefore this study fails to reject null hypothesis. In other word if similar study is conducted hypothetically 100 times from same sampling frame, 95 times the result will show a difference of proportions between −0.37 to +0.01. Similarly comparison of two means can be done by student t test, as described in Table-3Table 3

Comparing means two independent groups by using t test


Step 1	State H₀ and H₁	H₀: No difference in mean TSB in phenobarbitone and placebo groupsH₁: Less TSB in phenobarbitone group than in placebo group
Step 2	One or two sided test	One sided test
Step 3	Test used	t test Degree of freedom = (n₁+n₂−2) =73
Step 4	Select the level of significance	α = 0.01
Step 5	Determine critical value	For one sided α = 0.01, looking at reference table (available in standard textbooks of biostatistics), the value of t distribution that separates the lower 1% (α = 0.01 divided into 2) from rest 99% at degree of freedom of 73 will be −2.39. We will therefore reject the null hypothesis if the calculated t statistic is less than critical value of −2.39
Step 6^@	Calculate test statistic	t =Difference of means/Standard error of difference of means = mean₁-mean₂/Pooled standard deviation* square root [(1/n₁)+(1/n₂)] Calculations according to data given above will result in t statistic of −2.83
Step 7	Accept or reject null hypothesis	As calculated t statistic is less than −2.39 (Figure 1) we will reject the null hypothesis
Step 8	State the conclusion	Phenobarbitone reduces the mean serum bilirubin in the investigated sampling frame.

^@

Similarly the result can also be reported in terms of confidence interval. For above mentioned example confidence interval can be calculated as follows

For 95% CI value of Confidence coefficient is 1.96

Thus for above said data

As this 95% confidence limits is unidirectional and less than 1, this study rejects null hypothesis. In other word if similar study is conducted hypothetically 100 times from same sampling frame, 95 times the result will show a difference of mean TSB between −3.11 to −1.49.

Inferential statistics: introduction to hypothesis testing