Exploring, summarizing and presenting data. Berghold, IMI, MUG

Transcription

1 Exploring, summarizing and presenting data

2 Example Patient Nr Gender Age Weight Height PAVK-Grade W alking Distance Physical Functioning Scale Total Cholesterol Triglycerides 01 m II b m II b m II b f II b m IV f III f II b m IV m III f III m II b m II b f II b m II b m II b f IV f II b f II b m II b m III f III m II b f IV m III m II b

3 Scales Nominal scale Ordinal scale Numerical scale

4 Nominal Scale The values of any two study units can be classified either as identical or non identical hair colour place of birth blood group Binary (dichotomous) variables: gender, rhesus factor,...

5 Ordinal Scale Observation are still classified but some observations have "more" or are "greater than" other observations. school grades stage of breast cancer side effect of a drug (mild, average, severe) pain-scores...

6 Numerical Scale continuous (e.g. age, height - measurements) discrete (e.g. number of fractures, number of children - counts) weight body temperature blood pressure serum cholesterol...

7 Types of Data Qualitative data categorical variable Nominal scale Ordinal scale Quantitative data Discrete variables Continuous variables

8 Examples Protein measured in urine Spontaneous urine using test strips (neg., pos.: +,++,+++) 24 hours sample of urine protein g/24hours Smoking Consumed tobacco g/day Number of smoked cigarettes per day Non-smoker, smoker

9 Criteria - measurements Reliability Validity Ease of Use

10 Reliability reliable unreliable

11 Validity Valid Not valid

12 Descriptive Statistics Exploring and presenting data in form of graphs Summarizing - data reduction (mean, variance etc.) Presenting data in form of tables

13 Frequency Qualitative data absolute and relative frequency Quantitative data define class intervals Determine the number of class intervals There should be enough class intervals to show the shape of the distribution but not too many that minor fluctuations are noticeable.

14 Graphs Barchart Piechart Histogram Box-and-whisker plot Scatterplot Time series plot...

15 Barchart number of decayed teeth in pupils decayed teeth in pupils cumulative 30 frequencies percentage percentage ,3 33, ,7 68,0 9 12,0 80, ,3 89,3 2 2,7 92,0 4 5,3 97,3 1 1,3 98,7 absolute frequency ,3 100,0 total , number of decayed teeth in pupils

16 Piechart PAVK-Grade IV 24% II b 50% III 26%

17 Histogram and cumulative distribution 0,35 1,0 0,30 0,8 0,25 rel. frequency 0,20 0,15 F(x) 0,6 0,4 0,10 0,2 0,05 0,00 0,0 1-1,5 1, ,5 2, ,5 3, ,5 4, ,5 5, ,5 1, ,5 2, ,5 3, ,5 4, ,5 5,5-6 FT3 FT3

18 TRIGLYCERIDES (mg / 100 ml) Histogram 1 frequency Std.dev. = 38,83 Mean = 129 N = 80,00

19 Histogram TOTAL CHOLESTEROL (mg / 100 ml) frequency Std.dev. = 92,46 Mean = 220 N = 80,00

20 Histogram frequency Std.dev. = 21,97 Mean = N = 80,00 SYSTOLIC BLOOD PRESSURE (mmhg)

21 Types of Distribution a) unimodal b) skewed positively c) skewed negatively c) bimodal e) trapezoid f) truncated g) L- shaped h) J - shaped i) U - shaped

22 Scatterplot HDL LDL

23 Summarizing Data Common statistics used to summarize data and describe certain attributes of a set of data. Measures of location: the central tendency Measures of dispersion: the spread of data Mean Median, quantile Mode Variance, standard deviation Range Interquartile range

24 Mean Mean = arithmetic mean x = 1 n n i= 1 x i Note: The mean is sensitive to extreme values

25 Example Values: 1, 2, 30 x = ( ) 3 = 11 mean: x =

26 Variance, standard deviation s 2 = 1 n 1 n ( x ) i x i= 1 The variance of a data set is the arithmetic mean of the squared differences between the observations and the mean. s = s The standard deviation is primarily used to describe data. It is the square root of the variance. In many circumstances the large majority (about 95%) of a set of observations will be within two standard deviations of the mean (depends on the shape of the distribution normal distribution) normal range 2 2

27 Example The number of cows 4 farmers own in 3 villages village 1 village 2 village 3 observations 3, 6, 7, 4 5, 5, 5, 5 0, 0, 0, 20 mean x = 5 x = 5 x = 5 standard deviation s = 1.8 s = 0 s = 10.0

28 Time Series Plot R-TCI Induction of Anaesthesia Time Course (min) all data points: n = 30

29 Geometric mean Geometric mean The geometric mean is generally used with data measured on a logarithmic scale G = n x1x2... x n logg = n i= 1 log x n i The logarithm of the geometric mean is equal to the mean of the logarithms of the observations

30 Median Median The median is the central value of the distribution if n is odd ~ x = x n+ (( 1) / 2) if n is even ~ x 1 2 ( x + ) n x = n ( / 2) ( / 2+ 1)

31 Mean - Median Example: n = 3 values: 1, 2, 30 median ~ x = 2 : mean: x =

32 Skewness by mean, median and mode skewed negatively x < Me < Mo skewed positively Mo < Me < x

33 Quantiles The α-quantile The median is only a special case that is based on rank order. α-quantile x α : that at least α % of measurements are smaller or equal than the value x α. 1st quartile (α = 0.25) 2nd quartile or median 3rd quartile (α = 0.75) Percentiles (centiles)

34 Quantiles The α-quantile x α Calculation: α*n, rankorder m if α*n is not an integer, than m is the next integer following α*n and x α = x (m). if α*n is an integer, than m = α*n and x m + x 2 m+1 x α =

35 Quantiles

36 Quantiles Data: , 2, 2, 6, 7, 2, -40, 2, 3, 2, 1, 1, 12, 3, 4, 0-40, 0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 5, 6, 7, 12 Q 1 = 1.5 Me = 2.0 Q 3 = 4.5 Interquartile range = Q 3 Q 1 = 3

37 Interquartile Range Interquartile range The 50% central range is sometimes used to describe variability IQR = 3rd quartile - 1st quartile

38 Box-and-Whisker Plot maximum 3rd quartile median 1st quartile minimum

39 Example Box-and-Whisker Plot 6 one-second-capacity (L) Gender female 0 N = yrs 9-12 yrs yrs age groups male

40 In bunten Bildern wenig Klarheit, viel Irrtum und ein wenig Wahrheit J. W. v. Goethe

41 Presentation of Results Numerical Presentation Data summary should not be by the mean (median) alone, but some indication of variability should also be provided. E.g.: "... the mean diastolic blood pressure was mm Hg (SD 11.9)." mean: standard deviation: quote it to one extra decimal place compared with the raw data (depending on amount of data) display with same precision as mean or with one more decimal place.

42 Tables Mean (SD) Age 67,8 (10,8) Total Cholesterol 213,3 (41,1) Triglycerides 129,4 (72,0) frequency % Gender f 35 (46) m 41 (54) PAVK-Grade II b 38 (50) III 20 (26) IV 18 (24)