Research studies are designed and conducted with many different aims. Research question is the uncertainity about something in the population that investigator wants to resolve. A good research question should be relevant, novel, feasible and ethical. Hypothesis is the statement of a research question that summarizes key elements of the proposed study: sampled population, intervention or predictor, outcome variable and effect size. Hypothesis is needed for any study that will use tests of statistical significance to compare findings among two or more groups. A simple hypothesis contains one predictor/intervention and one outcome variable while a complex hypothesis contains more than one intervention or outcome variable. Hypothesis should be specific leaving no ambiguity and should be formulated in advance. It should precisely summarize nature and source of studied subjects and how outcome variables will be measured. For example Arya V B et al (<xref ref-type="bibr" rid="bib001">1</xref>) 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 onesided hypothesis, which specifies the direction of association between intervention and outcome variable i.e. reduction in neonatal hyperbilirubinemia with use of phenobarbitone. A two-sided hypothesis proposes only that an association exists; it does not specify the direction. One-sided hypothesis is appropriate when only one direction for an association is important or biologically plausible or there is good evidence from previous studies that an association is unlikely to occur in two directions. Advantage of choosing a one-sided hypothesis is obtaining the statistical significance with a smaller departure from hypothesized value. Investigator should make a choice between one-or two-sided hypothesis beforehand. Switching to one-sided hypothesis to reduce p value is not correct.A study may have several hypothesis and one may advance as many of them as makes sense. But it is prudent to specify one as the primary hypothesis. It would help to focus the study on its main objective besides providing a clear basis for sample size calculation.</sec><sec id="sec002"><title>Null and alternative hypothesis

The null hypothesis denoted by H_O is the formal basis for testing statistical significance. The null hypothesis is a statement claiming that there is no difference between proposed or hypothesized value and the population mean value. The alternative hypothesis denoted by H_A is a statement that disagrees with the null hypothesis. For example in the above mentioned study:

H_O: ”Oral phenobarbitone does not cause reduction in incidence of neonatal hyperbilirubinemia in term ‘high risk’ newborns”.

H_A: ”Oral phenobarbitone causes a 50% reduction in incidence of neonatal hyperbilirubinemia in term ‘high risk’ newborns”.

Assuming that null hypothesis is true statistical tests help to estimate the probability that an association (reduction in incidence of hyperbilirubinemia) observed in the study is due to chance. If the null hypothesis is rejected as a result of the sampled data then only the alternative hypothesis is accepted.

Type I and Type II errors

Two types of errors can be made while testing for statistical significance (Table 1). Although these errors can not be eliminated altogether, the investigator can reduce the likelihood by taking some steps at the time of designing the study, for example increasing the sample size. Table 1

Errors in hypothesis testing

		True situation

		Difference exists(H_A true)	No difference(H_O true)

Results of the study	Finds difference(Rejects H_O)	Situation ACorrect(Power of the study)	Situation BType I or α error
Results of the study	No difference(Does not reject H_O)	Situation CType II or β error	Situation DCorrect

Consider situation B in the Table 1. The null hypothesis is being rejected i.e. study is finding the difference (by chance) when in actual situation none exists. This is called α error or type I error. Converse is true in situation C when null hypothesis is not being rejected i.e. study is not finding the difference when in actual situation one exists. This is called â or Type II error. Power of the study (1-â) is the probability of rejecting the null hypothesis when it is indeed false. If â is set at 0.10, then the investigator has decided that he is willing to accept a 10% chance of missing an association. This is equal to a power of 0.90; that is 90% chance of finding an association. The ideal study is one in which the power is high. This means that the study has a high chance of finding a difference if one exists; and very importantly if the study demonstrates no difference between groups the researcher can be reasonably confident in concluding that none exists in reality. As a general rule higher power is achieved by increasing the sample size. It is important to be aware of this because all too often studies are reported that are simply too small to have adequate power to detect the hypothesized effect. In other words, even when a difference exists in reality it may be that too few study subjects have been recruited.

Sample size calculation

Researchers must learn how large a sample size is needed before beginning their research because they may not be able to determine significance when it occurs. Readers of the medical literature also need to know what sample size was needed in a study. This is especially true for a study with negative results. Institutional review boards and granting agencies also require sample size estimates before giving approval to the proposals. Calculation of an appropriate sample size for a study is based on calculation of three main factors: á or Type I error, Power and effect size of the intervention expected. These factors effect the sample size as outlined in Table 2. Table 2

Factors that determine sample size

Factor	Magnitude	Effect	Sample size

á or Type I error	Small	Difficult to achieve significance	↑
á or Type I error	Large	Significance easier to obtain	↓
Power	Low	Finding the true situation less likely	↓
Power	High	Finding the true situation more likely	↑
Effect size	Small	Difficult to identify	↑
Effect size	Large	Easier to identify	↓

Ideally, á error should be set to be zero and power should be set to be 100%, eliminating the possibility of false positive and false negative results. In general, for any given number of subjects there is a trade-off between Type I and Type II errors. Everything being equal, the more one is prepared to accept one kind of error, the less it will be to risk the other. In practice they are made as small as possible. á for most studies is set at 0.05 and power at 80% (â 0.20). These are arbitrary values and can be changed according to research question and availability of resources. In general one should use a low α when research question makes it particularly important to avoid a false positive result, such as testing the efficacy of a potentially dangerous medication. If it is important to avoid a false negative result a low α and low effect size should be used. For finding sample size in case outcome is a difference of means, one also needs a good estimate of standard deviation in the population. Once these factors have been established, there are computer programs and formulae available for calculating the required sample size. Certain outcomes and more complex study designs may require further information, and calculation of the required sample size is best left to someone with appropriate expertise. Methods for sample size calculation in two common situations are detailed in the following sections. Results of a sample size calculation are entirely dependent on estimates of effect, power and significance, as discussed above. Thus, a range of values should be incorporated into any good investigation in order to give a range of suitable sample sizes rather than a single ‘magic’ number.

Sample size calculation for a difference in means

Let’s consider an example to make the things simpler. A randomized trial was done by Reddy P S et al (2) to evaluate whether prophylactic use of cisapride will reduce the incidence of feed intolerance and gastro-esophageal reflux, and improve gastric emptying in early neonatal period in preterm babies. The first step in calculating the sample size would be to find the baseline variability in gastric emptying time. If the standard deviation is high say 40 minutes then a difference of <40 minutes found at the end of the study reflects a relatively small intervention effect relative to natural spread of data. To incorporate this factor in sample size calculation the concept of standardized difference needs to be incorporated 1

In practice target difference expected and standard deviation is unlikely to be known in advance, but one may get it from other similar studies from same population or from a pilot study. In above given example authors hypothesized a target difference of 30 minutes with standard deviation of 30 minutes. Therefore standardized difference to be detected, calculated using above mentioned equation was 30/30 = 1. Using this value sample size can be calculated using a nomogram (3) or a specific sample size formula. An appropriate formula for comparing means in two groups of equal size is as follows: 2

In equation (2) n is the number of subjects required in each group, d is the standardized difference and C is a constant defined by the values chosen for Power and á error. Some commonly used values for C are given in Table 3. Table 3

Commonly used values for C

	Power (%)

	50	80	90	95

á error
0.05	3.8	7.9	10.5	13.0
0.01	6.6	11.7	14.9	17.8

The number of participants required in each arm of the trial to detect a standardized difference of 1.0 with 90% power and α error of 0.05 is as follows

Thus 21 participants are required on each arm of the trial which agrees with the estimate provided by the authors.

Sample size calculation for a difference in proportions

For studies comparing proportions in two groups a different formula can be used: 3

In a recently published trial, Steer P et al (4) did a randomized controlled trail on use of high dose caffeine citrate for extubation of premature infants. They hypothesized a reduction of extubation failure rate from 32% to 16% using high dose caffeine citrate. Using this data and considering a power of 80% and á error of 0.05 let’s calculate sample size using formula (3).

Putting value of “c” from table 3 for power of 80% and á error of 0.05:

This sample size of 109 is in accord with author’s estimate of 111 in each group.

Special circumstances

In case of unequal sample sizes initial sample calculation is done assuming that the groups are unequal. Then this total sample size (N) can be adjusted according to the actual ratio of the two groups (k) 4

Individual sample sizes in each of the two groups are N’/(1+k) and kN’(1+k). Even in best designed and conducted studies some subjects may be lost in follow up or refuse to continue in the study. It is very important to account for this possible loss of subjects at the time of designing the study by adding a suitable percentage (10-20%) to the calculated sample size. We have tried to give a simplified explanation of sample size calculation in two most commonly used study designs. For more complicated study designs help of a statistician should be sought for.

Hypothesis and Sample Size Calculations: Basic Concepts