Things you need to know about the Normal Distribution. How to use your statistical calculator to calculate The mean The SD of a set of data points.

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1 Things you need to know about the Normal Distribution How to use your statistical calculator to calculate The mean The SD of a set of data points. The formula for the Variance (SD 2 ) The formula for the Standard Error of the Mean (SEM) The formula for the Coefficient of Variation (CV%) The formula for the Standard Normal Deviate (z)

2 Student s t Test A Paired Student s t Test which is simply calculated using the formula : (Mean of the Differences between the Paired Results). (SEM of the Mean of the Differences between the Paired Results) This is the only Student s t Test you can do yourself with just a statistical calculator. For the rest of the examples you need to use Excel.

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4 Student s t Test

5 Student s t Test

6 Student s t Test

7 Student s t Test To decide whether or not you should use the t-test : Two-Sample Assuming Equal Variances tool or the t-test : Two-Sample Assuming Unequal Variances you must carry out a F-Test. The F Test is also known as Fischer s F Test after the statistician who invented it. It is a very simple test you divide the greater variance by the lesser variance. Ideally the answer should be one. However depending on how many items you have in the two datasets determines how far away from one the answer can be before the two datasets are deemed to have significantly different variances. In the example just discussed - Placebo : Start versus New Drug : Start - the value of F is / = which is unlikely to be an indication of a significant difference. This is confirmed by the Excel F Test Two Sample for Variances, (See Figure 7)

8 R A Fisher s F Ratio Test Notice that Excel does not conform to the rule you divide the greater variance by the lesser variance nevertheless its internal tables for assessing the p-value of F are correct! So with a p value of 0.45 we are justified in using the t-test : Two-Sample Assuming Equal Variances

9 Things you need to know about the t Test and the F Ratio Test The formula for the Paired Student s t Test : (Mean of the Differences between the Paired Results). (SEM of the Mean of the Differences between the Paired Results) For a Paired Student s t Test the degrees of freedom is equal to the number of data rows minus one. For an Unpaired Student s t Test the degrees of freedom in a is equal to sum of the number of data rows in the two groups minus two. For Fisher s Variance Ratio Test you divide the greater variance by the lesser variance.

10 It is immediately obvious from Figure 10 that the datasets are no way near fitting the characteristic bell shaped Normal Distribution curve and the main reason for this is the paucity of data points. There is, however, an impression that the Coeliac data are clustering lower than the Crohn s disease data. Is this statistically significant? Two statisticians Henry Mann and Donald Whitney developed a statistical test now known as the Mann-Whitney U Test to analyse these type of datasets. Essentially the theory is

11 If you have two columns of data and you pool them, then sort them into ascending order, then add up the ranks for the items that came from the first column then add up the ranks for the items that came from the second column and compare that with the sums of the ranks of the items that came from the second column then compare the these two sums if they (the two sums) are about the same then the two columns of data are probably closely overlapped. They modelled this concept for all types of overlapping, non-overlapping and unequal columns of data and came up with this U statistic that they could assign a p significance value to. There is an online calculator at When you put these data into their calculator then you get the following answer (see Figure 11).

12 The Mann Whitney U-Test

13 The Mann Whitney U-Test Things you need to know about the Mann Whitney U-Test This is a non-parametric statistical test based upon the comparison of the relative rankings of two columns of data. If the data are closely interleaved then you will not get a significant U-Test result. If the members of one dataset predominate in the lower overall rankings than the other then there is a chance that you will get a significant U-Test result There is no U-Test in Excel so use an online calculator

14 MEDIANS and QUARTILES Another way of looking at a distribution of data is to sort the items into an ascending order and then divide the series into four equal parts. These parts are called quartiles. For example if you have 100 data points you sort them and then divide them into into four groups of 25. The FIRST QUARTILE lies between the 25 th and 26 th values The SECOND QUARTILE lies between the 50 th and 51 st values The THIRD QUARTILE lies between the 75 th and 76 th values The SECOND QUARTILE is also known as the MEDIAN. You should also remember that The FIRST QUARTILE is identical to the 25 th percentile. The SECOND QUARTILE is identical to the 50 th percentile. The THIRD QUARTILE is identical to the 75 th percentile. If you have an odd number of data points then the median is easy to determine it is the (0.5 + n/2) item in the sorted dataset, eg. if we have 101 data points then the median is at the ( /2 ) = 51st item in the sorted list.

15 MEDIANS and QUARTILES In this series the median is easy to determine it is the ( /2) = 9th item; ie 195. The first and third quartiles fall between the 4 th and 5 th values (155 and 160) and the 13 th and 14 th values (250 and 300). Simple linear interpolation would give a fairly good estimate of the first quartile because the 4 th and 5 th values are numerically very close to each other.. ( )/2 = Excel gives an answer of 160. In contrast simple linear interpolation between the 13 th and 14 th data points gives a value of 275 whereas Excel gives a value of 250. So it is always best to use the formulae and procedure that Excel uses. (You will not be expected to do this in an exam.) In the lecture we will use an Excel template that has been set up for you to determine the quartiles and vertical Box and Whisker Plots for up to six variables by 100 points per variable, (see Figure 14)

16 Box and Whisker Plot A convenient way to visualise the Quartiles in a dataset

17 Things you need to know about Medians and Quartiles The definition of a Median The definition of a Quartile How to calculate the Median from a series of data values

18 THE CONSORT STATEMENT CONSORT is the acronym for Consolidated Standards of Reporting Trials and is the pro-forma for the design of and data collection from randomised controlled trials. These are trials that test the effectiveness of a clinical intervention in a defined group of patients or a defined combined group of patients and normal controls. An article that describes the 2010 version of CONSORT is in Appendix 2 to these notes.

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20 THE STARD STATEMENT STARD is the acronym for Standards for Reporting of Diagnostic Accuracy and is the pro forma for the design and data collection of trials where the accuracy of new diagnostic investigation e.g. a laboratory test, is compared to that of either current usual practice or an index diagnostic investigation. By index is meant the best available diagnostic investigation for the clinical condition being considered. A copy of the STARD statement is in Appendix 3.

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22 POWER OF A STUDY AND NUMBER NEEDED TO ACHIEVE THAT POWER DESIRABLE POWER = 0.8 OR 80% N IS BACK CALCULATED FROM THE FORMULA OF THE STATISTICAL TEST YOU HAVE CHOSEN TO USE IN YOUR STUDY.. Very often this is a Student s t Test Do not need to calculate manually use a program or on-line calculator PS : Power and Sample Size Calculator which can be downloaded from

23 Sample size for paired data: n 2 ( Z Z d /2 2 difference ) 2 where : n sample size standard deviation of diffference clinically Z Z meaningful corresponds to power (.84 the within - pair difference difference 80% power) / 2 corresponds to two- tailed significan ce level (1.96 for.05)

24 Sample size for unpaired data: n 1 2 ( r 1) ( Z Z /2 2 r difference ) 2 where : n r ratio of standard deviation of diffference clinically meaningful difference in means of the outcome Z Z 1 size of / 2 smaller group larger group corresponds to power (.84 to smaller group the characteristic 80% power) corresponds to two- tailed significan ce level (1.96 for.05)

25 If you are unsure of what shape your data distribution is going to be then you can use the formulae for the estimation of the errors above and below your expected median; (these are taken from the book by Snedecor). You round down the lower limit and round up the upper limit. These two equations actually give you the item number of the data point that corresponds to the limit so to use them you need to have your data sorted into ascending order.

26 Things you need to know about the Design of Research Projects The CONSORT STATEMENT definition The STARD STATEMENT definition Studies need to have sufficient numbers of participants in them to achieve a POWER OF 0.8 (80%) The formulae for calculating the index numbers of the Upper and Lower Limits of a Median

27 5.3 THE SIMULATED PROJECT DETAILS Behind any worthwhile project there needs to be a scientific basis. Currently there is significant interest in a group of biomolecules that are grouped under the name advanced glycation end-products or AGE components. It s an acronym that sits well with the biological process under which they are formed the longer glucose and proteins remain in contact with each other the further to the right the reaction shown below goes: glucose + proteins glycated proteins It has been shown that the lysine residue in proteins are particularly prone to react with the open form of glucose (the keto form) and although this is initially reversible, there are molecular re-arrangements (especially the Amadori re-arrangement) which leads to a stable adduct. As a group these stable adducts are referred to as advanced glycation end-products. In mammalian systems these AGE compounds appear in the blood and tissues of diabetic individuals because of their tendency to have prolonged episodes of high glucose concentrations in their blood and tissues. This simulated project therefore has this as its scientific basis: Levels of glycated proteins are elevated in patients with high average blood glucoses. Individuals with high BMI s are prone to Type 2 diabetes.

28 This simulated project therefore has this as its scientific basis: Levels of glycated proteins are elevated in patients with high average blood glucoses. Individuals with high BMI s are prone to Type 2 diabetes. A high BMI in an individual with a tendency towards Type 2 diabetes will have elevated glycated protein levels. The sooner the individual is advised of their serum glycated protein level, the sooner they and their doctor can plan strategies to lower the concentration of it by managing their weight and blood glucose. Glycated protein concentrations usually require the blood sample to be collected and sent to the lab. The result is usually not available until the next day. This means that the doctor cannot start treating a patient with a high glycated protein level until they see them at their next appointment. This delay is a threat to the patient s health. A new point of care (POC) analyser has just been released which produces the result in 5 minutes. This means that while the patient is still with the doctor, they can get a result and start treatment straight away if the glycated protein measured on the POC analyser is elevated.

29 The clinic doctors and the pathology laboratory therefore decided to collaborate on a study of the effects of this new POC analyser on their patient management. To be objective about this, they decided that they would do the POC test on all patients but that only half of those POC tests would be used for management decisions. They would also send samples on all patients as usual to the lab for analysis. They also decided that they would like to know if the POC result plus immediate management had a particularly beneficial outcome(s) for their high risk patients. They defined high risk patients as those with a BMI of greater than 27. (BMI = body mass index = body weight in kgs divided by their height in metres squared).

30 342 patients came to the clinics during the study period. Of these 116 were ineligible for inclusion in the study which meant that 226 were eligible. Of those 226 patients, 92 were eligible to go into the BMI >27 group. These 92 patients were alternately allocated to the POC Result Withheld group or the POC Result Used Immediately group. It turned out that of the 46 that went into the POC Result Withheld group, only 45 of them turned up for an identical review at 6 months. Of the 46 that went into the POC Result Used Immediately group, only 44 of them turned up for review at 6 months. There were 134 patients in the BMI <=27 group and 67 were sequentially allocated to the POC Results Used Immediately and 67 allocated to the POC Result Withheld group. Of the 67 that were in the POC Result Withheld group, 10 were lost to follow up at 6 months. In comparison 15 were lost to the 6 month follow up in the POC Result Used Immediately group.

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32 So in summary, the patients included in the statistical data analysis were: 45 in the high BMI, POC Result Withheld group. 44 in the high BMI, POC Result Used Immediately group. 57 in the <=27 BMI, POC Result Withheld group. 52 in the <=27 BMI, POC Result Used Immediately group.

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