124 Emerg Med J 2001;18:124–130 AN INTRODUCTION TO STATISTICS Emerg Med J: first published as 10.1136/emj.18.2.124 on 1 March 2001. Downloaded from Article 6. An introduction to hypothesis testing. Parametric comparison of two groups—1 P Driscoll, F Lecky Objectives Table 1 Types of test of significance for two group x Dealing with paired parametric data comparison x Comparing confidence intervals and p Parametric Non-parametric values In covering these objectives the following terms Quantitative Nominal z test ÷2 test will be introduced: Paired Fisher exact test x Parametric and non-parametric analysis Independent Ordinal x Paired z test t test Wilcoxon rank sum test Paired Mann-Whitney x Paired t test Independent Kolmogorov-Smirnov We have shown previously that statistical infer- ence enables general conclusions to be drawn from specific data. For example estimating a distribution of a population these features are population’s mean from a sample mean. At known as parameters. Parametric analysis relies first glance this may not appear important. In on the data being normally (or nearly) distrib- practice however the ability to make these esti- uted so that an estimation of the underlying 3 mations is fundamental to most medical inves- population’s parameters can be made. These tigations. These tend to concentrate on dealing can then be used to test the null hypothesis. As with one or more of the following questions: only quantitative data can have a normal distri- Have the observations changed with time bution, it follows that parametric analysis can and/or intervention? only be used on quantitative data (table 1). Do two or more groups of observations http://emj.bmj.com/ diVer from each other? Key point Is there an association between diVerent All parametric tests use quantitative data observations? but not all quantitative data have to be ana- To answer these questions many diVerent lysed using parametric tests. types of statistical inference tests have been developed to deal with varying sample sizes and diVerent types of data. Though the tests NON-PARAMETRIC ANALYSIS diVer they have the common aim of assessing These tests of the null hypothesis do not on September 27, 2021 by guest. Protected copyright. whether the null hypothesis is likely to be cor- assume any particular distribution for the data. rect (box 1). They are known collectively as Instead they look at the category or rank order “tests of significance”.1 of the values and ignore the absolute diVerence between them. Consequently non-parametric analysis is used on nominal and ordinal data as Box 1 The null hypothesis well as quantitative data that are not normally There is no diVerence between the groups (or nearly normally) distributed (table 1). with respect to the measurement made. IfadiVerence exists between the study groups, it is more likely to be found using para- metric tests. It is therefore important to know The significance test chosen is dependent upon for certain if the data are normally distributed. the type of data we are dealing with, whether it You can sometimes determine this by checking has a normal distribution and the type of ques- the distribution curve of the plotted data. A tion being asked.2 Once the distribution of the more formal way is to use a computer to show data is known, you can tell if the null how precisely the data fit with a normal distri- hypothesis should be tested using parametric Accident and bution. This will be described in greater detail or non-parametric methods. Emergency in the next article. When data are not normally Department, Hope distributed attempts are often made to trans- Hospital, Salford M6 8HD, UK Parametric and non-parametric analysis form it so that parametric analysis can be car- PARAMETRIC ANALYSIS ried out. The commonest method used is loga- Correspondence to: A normal distribution is a regular shape. As such rithmic transformation.2 This has the added Mr Driscoll, Consultant in it is possible to draw the curve exactly by simply advantage of allowing geometric means and Accident and Emergency Medicine (pdriscoll@ knowing the mean, standard deviation and vari- confidence intervals to be calculated that have hope.srht.nwest.nhs.uk) ance of the data. When considering a normal the same units as the original data. www.emjonline.com An introduction to hypothesis testing 125 approximately normally distributed.4 As this is Key points often the case the paired t test, rather than its z Emerg Med J: first published as 10.1136/emj.18.2.124 on 1 March 2001. Downloaded from x Non-parametric analysis can be used on counterpart, is more commonly seen in the any data but parametric analysis can only medical literature. To show how these tests are be used when the data are normally applied consider the following examples. distributed. x Provided they are appropriately used, parametric tests derive more information PAIRED Z TEST about the whole population than non- Dr Egbert Everard continues to work in the parametric ones. Emergency Department of Deathstar General. His consultant, Dr Canute, asks him to find out if Neverwheeze, the new bronchodilator for Having determined that parametric analysis is asthmatics, significantly changes patient’s peak appropriate you then need to select the best flow rate (PFR). To do this he follows the sys- statistical test. This depends upon the size of tematic approach described in box 2. the samples and the type of question being asked. In all cases however the following 1 State the null hypothesis and alternative systematic approach is used (box 2). hypothesis of the study Having considered the problem, Egbert writes Box 2 System for statistical down the null hypothesis as: comparison of two groups “There is no diVerence in asthmatic patient’s x State the null hypothesis and the alterna- PFR before and after receiving Never- tive hypothesis of the study wheeze”. x Select the level of significance This can be summarised to: x Establish the critical values Mean diVerence in PFR = 0 x Select a sample and calculate its mean The alternative hypothesis is the logical oppo- and standard error of the mean (SEM)* site of this, that is: x Calculate the test statistic “There is a diVerence in asthmatic patient’s x Compare the calculated test statistic with PFR before and after receiving Never- the critical values wheeze”. x Express the chances of obtaining results This can be summarised to: ≠ at least this extreme if the null hypothesis Mean diVerence in PFR 0 is true *or estimated standard error of the mean 2 Select the level of significance (ESEM) if using a sample size < 100.4 If the null hypothesis is correct the PFR before and after Neverwheeze should be the same. However, even if this were true it would be very unlikely they would be exactly the same http://emj.bmj.com/ Paired or independent parametric because of random variation between patients. analysis You would expect some PFR to increase and Two types of parametric statistical tests can be others to fall. Overall however the mean diVer- used to compare the means of two study groups. ence between the two groups should be zero if The choice depends upon whether the data you the null hypothesis is valid. are dealing with are independent or paired. When groups are widely separated it is likely Data can be considered to be paired when that the null hypothesis is not valid. For exam- two related observations are taken with analysis ple, if 100 people died in your department one on September 27, 2021 by guest. Protected copyright. concentrating on the diVerence between the day and none the next then its highly unlikely paired scores. Examples of these type of data that a diVerence this big would be attributable include: to chance. In contrast you would be less confi- x “Before” and “after” studies carried out on dent to rule out the eVect of chance if the dif- the same subjects ference in death rate was only 1. The question x Observations made on individually matched therefore is how big does the diVerence need to pairs where only the factor under investiga- be before the null hypothesis can be rejected? tion is diVerent Rather than guessing, it is better to consider Independent data are when the subjects for what values are possible. If we measured the the two groups are picked at random such that mean PFR diVerence in groups of asthmatic selection for one group will not eVect the sub- patients selected randomly we would find the jects chosen for the other. The tests used to sample means form a normal distribution analyse these data will be discussed in the next around the population’s mean diVerence. As it article. For now we will concentrate on paired is a normal distribution it is possible to convert data. this to a standard normal distribution.5 The probability of getting a particular sample mean PAIRED TESTS can then be read from the table of z statistics Both z and t tests can be used to investigate the where: diVerence between the means of two sets of z = [sample mean diVerence −population paired observations. The former is, however, mean diVerence (µ)]/standard error of the only valid if the samples are suYciently large.4 mean (SEM) When this is not the case we can use the t sta- Where: tistic provided that the population of the mean SEM = population standard deviation (ó)/ diVerences in scores between the pairs is 'number in the sample (n) www.emjonline.com 126 Driscoll, Lecky selected. The z statistic derived from the sam- In this case we do not know the value of ó.