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Data Handling II: Describing and Depicting your Data Dr Yanzhong Wang Lecturer in Medical Statistics Division of Health and Social Care Research King's College London Drug Development Statistics & Data Management

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2 Types of data Quantitative data – continuous, discrete – distributions may symmetric or skewed Qualitative (categorical) data – binary – nominal, ordinal

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3 Long tail to leftLong tail to right Skewed Distributions

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Symmetric Distribution

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5 Summary statistics ‘Where the data are’ - location – mean, median, mode, geometric mean Used to describe baseline data and main outcomes ‘How variable the data are’ - spread – standard deviation, variance, range, interquartile range, 95% range Needed (primarily) to describe baseline data in RCT and cohort study

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6 Definition of the Mean The mean of a sample of values is the arithmetic average and is determined by dividing the sum of the values by the number of the values.

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7 Definition of the Median The median is the middle value. not affected by skewness and outliers, but less precise than mean theoretically.

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Ordered Blood Glucose Values

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Definition of the Mode The mode is the most frequent value. 9

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Ordered Blood Glucose Values 10

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Blood glucose (mmol/litre) Count Arithmetic Mean - outlier prone Mode - not necessarily central (categorical data) Median - only uses relative magnitudes Location = Central Tendency 11

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Relation of mean, median and mode If distribution is unimodal (has only one mode) then: Mean=median=mode for symmetric distribution. Mean>median>mode for positively skewed distribution. Mean

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Serum Triglyceride Levels Count Serum Triglyceride Levels from Cord Blood of 282 Babies 13

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log(Serum Triglyceride) Levels count Log(Serum Triglyceride Levels) from Cord Blood of 282 Babies 14

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Definition of the Geometric Mean The geometric mean of a sample of n values is determined by multiplying all the values together and taking the nth root (for only two values this is the more familiar square root). 15

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Geometric Mean A common example of when the geometric mean is the correct choice average is when averaging growth rates. Another Method: Take log of each value, find arithmetic mean and anti-log the result. Exp( (log(0.15) + … + log(1.66) )/40) = 0.467

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Serum Triglyceride Levels Count Mean=0.506 Median=0.460 Geometric Mean=0.467 Serum Triglyceride Levels from Cord Blood of 282 Babies 17

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Why measures of variability are important Production of Aspirin New production process of 100 mg tabs Random sample from process – mgs - mean 99 mg Random sample from old process – mgs - mean 99 mg Same means but new is better because less variable 18

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Definition of Range The range of a sample of values is the largest value minus the smallest value. New process the range is =5 Old process the range is =22 Range is simple ….. BUT – Only uses min and max – Gets larger as sample size increases 19

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Definition of Inter-quartile Range The inter-quartile range of a sample of values is the difference between the upper and lower quartiles. The lower quartile is the value which is greater than ¼ of the sample and less than ¾ of the sample. Conversely, the upper quartile is the value which is greater than ¾ of the sample and less than ¼ of the sample. 20

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Ordered Blood Glucose Values /4 of 40 = 10 3/4 of 40 = 30 21

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Blood glucose (mmol/litre) Count Inter-Quartile Range Lower quartile Upper quartile Inter-quartile range 22

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Standard deviation Neither measure uses the numerical values - only relative magnitudes A measure accounting for the values is the standard deviation Consider the aspirin data from the new process (mean 99 mg) Determine deviations from mean Square, add, average and square-root 23

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Measures of scatter/dispersion – ‘how variable the data are’ Range – smallest to biggest value – increases with sample size Standard deviation – measure of variation around the mean – affected by skewness and outliers Variance = square of standard deviation Interquartile range (IQR) – from 25th centile to 75th centile 24

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Plotting Data Histograms Stem and Leaf Plots Box Plots Stem Leaf Multiply Stem.Leaf by 10** Blood glucose (mmol/litre) 25

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Mean and standard deviation Best description if distribution reasonably symmetric (and single mode) Give full description if data have Normal distribution 26

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Mean 3, s.d. 1 Mean 5, s.d. 1 Mean 5, s.d. 2 27

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Properties of Normal distribution Symmetric distribution – mean, median and mode equal Completely specified by mean and standard deviation 95% of distribution contained within mean 1.96 standard deviations 68% within mean 1 standard deviation 28

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Continuous data, not Normally distributed If symmetric use mean and standard deviation If skewed use median and IQR Unless Positively skewed, but log transformation creates symmetric distribution – use geometric mean 29

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Nominal categorical data Mode. % in each category, especially when binary. Wheeze in last 12 months Frequency (n)% No Yes Total

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Ordinal categorical data Median and IQR if enough separate values. Otherwise as for nominal. 31

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Discrete quantitative data As for continuous data if many values, as for ordinal data if fewer.

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33 Difference Between Standard Deviation & Standard Error

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34 Measure of Variability of the Sample Mean Range, inter-quartile range and standard deviation relate to population (sample) not mean. To understand the difference carry out a sampling experiment using the Ritchie Index values

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35 Values of the Ritchie Index (Measure of Joint Stiffness) in 50 Untreated Patients Mean = (14+…+21)/50 = 12.18

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Values of the Ritchie Index Arithmetic Mean - outlier prone Median - only uses relative magnitudes Mode - not necessarily central (categgorical data) Location = Central Tendency

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37 Sampling Experiment Take a random sample (10) from the 50 values Determine the mean of the 10 values Repeat 50 times These means show variation - HOW LARGE IS IT ?

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38 Variations in Samples Values of the Ritchie Index Values of the Ritchie Index Values of the Ritchie Index Values of the Ritchie Index Values of the Ritchie Index Mean=12.18 Mean=10.00 Mean=12.60 Mean=13.40 Mean=11.50

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39 Ritchie Values Values of the Ritchie Index Original values (mean ; sd )

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40 Ritchie Values Sampling Experiment – Sample Means Values of the Ritchie Index Sample means (mean ; sd ) Original values (mean ; sd )

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41 Definition of the Standard Error The standard deviation of the sampling distribution of the mean is called the standard error of the mean.

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42 Increasing Sample Size Increased precision (smaller standard error) Less skewness Values of the Ritchie Index Sample means (mean ; sd ) Values of the Ritchie Index Sample means (mean ; sd ) n=10 n=15

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43 Standard error of the mean as a function of the sample size Sample Size Standard Error of the Mean

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44 Population of Gene Lengths n=20, Gene Length (# of nucleotides) Frequency

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45 Samples of size : n= Gene Length (# of nucleotides) Frequency

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46 Practical Confusion A mean is often reported in medical papers as 1.37 what is 1.37 ? sd or se ?

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Thanks! Tea break

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