CTRC Core Curriculum Seminar Series

Transcription

1 CTRC Core Curriculum Seminar Series Descriptive Statistics: Data Types and Measures, Central Tendency, Variability Chang-Xing Ma, PhD Associate Professor Department of Biostatistics, UB January 4, 2012

2 Disclosure Statement Chang-Xing Ma, PhD Nothing to disclose

3 Goals and Objectives Goals: Gain the knowledge of basic statistics and how to describe the data Objectives: Describe the data type Summarize data Understand Measure of Central Tendency Understand Measure of Dispersion

4 Outline Basic concepts of biostatistics Data type Summarize data Measure of Central Tendency Measure of Dispersion

5 Some terminology Statistics is the study of how to collect, organize, analyze, and interpret numerical information from data Biostatistics the theory and techniques for collecting, describing, analyzing, and interpreting health data.

6 Some terminology Population refer to all measurements or observations of interest Sample is simply a part of the population. But the sample MUST represent the population. A random sample is such a representative sample The sample must be large enough The sample should be selected randomly

7 Some terminology Parameter is some numerical or nominal characteristic of a population A parameter is constant, e.g. mean of a population Usually unknown Statistic is some numerical or nominal characteristic of a sample. We use statistic as an estimate of a parameter of the population It tends to differ from one sample to another We also use statistic to test hypothesis

8 Population: all U.S. persons ~ Normal (µ h,σ h2 ), Parameters (µ w,σ w2 ), A random sample: sample size = 100 Generate Gender Height Weight A sample mean height: std height: mean weight std weight % of male (=1) statistics True Paramete

9 Sources of data Records Surveys Experiments Comprehensive Sample

10 Types of variables Quantitative variables Qualitative variables Quantitative continuous Qualitative nominal Quantitative discrete Qualitative ordinal

11 Data Types Numerical (Quantitative) numerical measurement Height Weight Categorical (Qualitative) with no natural sense of ordering Gender Hair color Blood type

12 Numerical Variable Continuous Range of values Discrete Height in inch Limited possible values Age - # of smoking per day # of children in a family

13 Determining Data Types Ordinal (Categorical) vs. Discrete (Numerical) Ordinal Cancer Stage I, II, III, IV Stage II 2 times Stage I Categories could also be A, B, C, D Discrete # of children: 0, 1, 2, 4 children = 2 times 2 children

14 Descriptive Statistics reducing a complex mass of data to a manageable set of information Descriptive Statistics: the summary and presentation of data to: simplify the data enable meaning full interpretation support decision making Numerical descriptive measures (few numbers) Graphical presentations

15 Inferential statistics From a sample to estimate population parameters to test hypothesis to build the model to reflect the population

16 The student test score (FCAT) Student ID Race Sex Reading Math Poverty W F W F W F W F B F B F W F W M W M W M W M W M B M B M H F B F W M W M Code: Race: W White B Black H Hispanic A Asian Sex: F Female M Male Poverty: 0 not poor 1 poor Problem 1 1.Among the 6 variables, which ones are qualitative and which ones are quantitative? 2.Is Race nominal or ordinal?

17 Descriptive Statistics Categorical variables: Frequency distribution Bar chart, pie chart Contingency tables Continuous variables: Grouped frequency table Central Tendency Variability

18 Simple Frequency Distribution An ordered arrangement that shows the frequency of each level of a variable. race Frequency Percent A B H W sex Frequency Percent F M

19 Simple Frequency Distribution It is useful for categorical variable For continuous variable, it allows you to pick up at a glance some valuable information, such as highest, lowest value. ascertain the general shape or form of the distribution make an informed guess about central tendency values

20 Bar Chart Race BY None summarizing a set of categorical data - nominal or ordinal data It displays the data using a number of rectangles, each of which represents a particular category. The length of each rectangle is proportional to the number of cases in the category it represents can be displayed horizontally or vertically they are usually drawn with a gap between the bars Bars for multiple (usually two) variables can be drawn together to see the relationship A B H W Race Horizontally

21 Pie Chart summarizing a set of categorical data - nominal or ordinal data It is a circle which is divided into segments. Each segment represents a particular category. The area of each segment is proportional to the number of cases in that category. Female Male Race PieExploded

22 Complex frequency distribution Table Distribution of 20 lung cancer patients at the chest department of Alexandria hospital and 40 controls in May 2008 according to smoking Smoking Cases Lung cancer Control Total No. % No. % No. % Smoker 15 75% 8 20% Non smoker 5 25% 32 80% Total

23 How about continuous variables? How data is distributed? Measure of Central Tendency Measure of Variability

24 DATA: Grouped Frequency Distribution for continuous variable Frequency Table Example Data [142.5, 157.5) 1.6% [157.5, 172.5) % [172.5, 187.5) % [187.5, 202.5) % [202.5, 217.5) % [217.5, 232.5) % Interval Size: New Data N: µ: σ:

25 Grouped Frequency Distribution BUT the problem is that so much information is presented that it is difficult to discern what the data is really like, or to "cognitively digest" the data. the simple frequency distribution usually need to condense even more. It is possible to lose information (precision) about the data to gain understanding about distributions. This is the function of grouping data into equal-sized intervals called class intervals. The grouped frequency distribution is further presented as Frequency Polygons, Histograms, Bar Charts, Pie Charts.

26 Describing Distributions Bell-Shaped Distribution 0.40 Normal distribution N (µ=0, σ 2 =1) t-distribution

27 Describing Distributions Skewed Distribution positively skewed distribution

28 Describing Distributions Skewed Distribution negatively skewed distribution

29 Describing Distributions Other Shapes Rectangular Bimodal

30 Other Shapes J-curve Describing Distributions

31 Probability density function - Normal z-transform green curve is standard normal distribution

32 Measure of Central Tendency Mean, Median, Mode The Mean average value not robust to outlying value X N i 1 N X i Length of hospital stays: 6, 4, 5, 9, 10, 7, 1, 4, 3, 4 Mean=( )/10=5.3

33 Measure of Central Tendency Mean, Median, Mode The Median is the point that divides a distribution of data into two equal parts robust to outlying value Length of hospital stays: sort data median=4.5 Split Data

34 Measure of Central Tendency Mean, Median, Mode The Mode is the midpoint of the interval that has highest frequency robust to outlying value, but sometimes misleading Length of hospital stays: sort data Mode=4, which occurred 3 times. Most frequently

35 Comparison between mean and median Mean Median

36 Comparison between mean and median Median Mean

37 Comparison between mean and median Mean Median

38 Summary Frequency distribution Histogram, Polygon graph Bar Chart, Pie Chart Describing Distributions Mean, Median, Mode DATASET:

39 Problem 2 In a study, we collected a medical measurements X for 4 patients Data of X: 2, 3, 5, 6 Mean of X? Median of X? Mode of?

40 The sample range Interquartile range Descriptive Statistics Variability The sample standard deviation (SD), variance Standard error of mean (SEM)

41 Measures of Dispersion - Range Range the difference between the lowest and highest For example, Age of Patients (years): lowest 2, highest 17 Range=2-17 years When sample size increases, the range tends to increase as well. (not robust)

42 Measures of Dispersion - Range All of curves have the same range Mean? Median?

43 Measures of Dispersion Percentiles, Deciles, Quartiles Percentiles: based on dividing a sample or population into 100 equal parts. Deciles divide the distribution into 10 parts Quartiles divide the distribution into 4 equal parts. 1 st quartile includes the lowest 25% of the values (Q1) 2 st quartile includes the values from 26 percentile through 50 percentile (Q2) - median 3 st quartile includes the values from 51 percentile through 75 percentile (Q3)

44 Measures of Dispersion Interquarile Range Interquarile Range the 25 percentile (1 st quartile) to 75 percentile (3 rd quartile) Age of Patients (years): st quartile 6, 2 nd quartile 8.5, 3 rd 13 Interquarile Range = 6-13 years Interquarile Range is a robust estimate of data variability

45 Measures of Dispersion Interquarile Range Robust estimate, less efficient

46 Deviations from the mean Variance and Standard Deviation deviation: observation - mean sum of deviation BUT ( x x) i ( x) x i 0

47 Deviations from the mean Variance and Standard Deviation Measure of how different the values in a set of numbers are from each other Variance: Standard Deviation: 2 2 ) ( 1 1 x x n s i 2 ) ( 1 1 x x n s i

48 Deviations from the mean Variance and Standard Deviation Data set: 2,3,5,6 2 1 Calculation: s ( xi n 1 x) 2 x x / n ( ) / 4 i 4.0 x Value of X (X- ) (X- ) =0 =10 Variance Standard Deviation s ( x ) n 1 x i x 10 /(4 1) 1 2 s ( x ) n 1 x i

49 0.60 Three normal distributions: mean=0 s 2 =1 s 2 =2 s 2 =0.5 Leptokurtic Homogenous Narrow scatter Mesokurtic Platykurtic Heterogeneous wide scatter Central Tendency mean=0 0,1 0,2 0,0.5

50 Example 2: FEV1 (litres) of 57 male medical students Table: FEV1 (litres) of 57 male medical students

51 Frequency Example 2: FEV1 (litres) of 57 male medical students Mean: 4.06 Variance: 0.45 SD: 0.67 Q1: 3.54 Q2 (Median): 4.10 Q3: Percentile 5.16 Range: 2.85 to FEV1 (litre)

52 The Meaning of Standard Deviation How the data are dispersed around mean Mean ± 1 SD represent 68.3% of the population Mean ± 2 SD represent 96% of the population Mean ± 3 SD represent 99.7% of the population

53 The Meaning of Standard Deviation ±SD % of Pop % 34% 1SD 1SD 2SD 48% 2SD 48%

54 Standard Error of Mean (SEM) How confident can we be that the sample mean represents the population mean µ? SEM=SD/ n SEM must be much smaller than the SD mean ± 1.96*SD cover 95% of the data mean ± 1.96*SEM cover 95% of the population mean SEM and SD are different!

55 Standard Error of Mean (SEM) Describing the scatter or spread of data, use SD Estimate population parameters, use SEM Epidemiologic study, SEM Clinical or laboratory research, SD

56 Put DATA below: None RUN Summarizing Data - Calculator Interval Size: Mean: 4.06 Variance: 0.45 SD: 0.67 Q1: 3.54 Q2 (Median): 4.10 Q3: Percentile 5.16 Range: 2.85 to New Data N: µ: σ: 100 Race 10 Ylim: 2,6 ReDraw

57 Box-Plot The box itself contains the middle 50% of the data. The upper edge (hinge) of the box indicates the 75th percentile of the data set, and the lower hinge indicates the 25th percentile. The range of the middle two quartiles is known as the inter-quartile range. The line in the box indicates the median value of the data. The + indicate mean value The ends of the vertical lines or "whiskers" indicate the minimum and maximum data values, unless outliers are present in which case the whiskers extend to a maximum of 1.5 times the inter-quartile range. The points outside the ends of the whiskers are outliers or suspected outliers

58 Box Plot Example 2 FEV1 of 57 students Serum triglyceride measurements in cord blood from 282 babies

59 What you can get from a box-plot? Graphically display a variable's location and spread at a glance. [Q1, Q2 (median), Q3, interquartile range] Provide some indication of the data's symmetry and skewness. Unlike many other methods of data display, boxplots show outliers. By using a boxplot for each categorical variable side-by-side on the same graph, one quickly can compare data sets. One drawback of boxplots is that they tend to emphasize the tails of a distribution, which are the least certain points in the data set. They also hide many of the details of the distribution. Displaying histogram in conjunction with the boxplot helps

60 frequency frequency Transformations triglyceride triglyceride log(triglyceride) LOG (triglyceride)

61 Summarizing data Univariate categorical variable Frequency distributions Bar Chart, Pie Chart

62 Summarizing data Univariate continuous variable Grouped frequency distributions Polygon or histogram Mean, Median, Mode, Percentile, Q1, Q2, Q3, extreme values Standard deviation, variance, range, interquartile range Box-Plot Normality test statistics

63 Next lecture ( Lecture 2) Bivariate one is categorical and the other is continuous variable t-test ANOVA

64 Lecture 3 categorical data analysis Bivariate both are categorical Contingency tables Chi-square test Response is categorical, predictors could be both types. Logistical regression

65 Lecture 4 Continuous response Correlation Multiple linear regression

66 Thanks. Question?