Basic statistical concepts

1. Basic statistical concepts xxia@uottawa.ca http://dambe.bio.uottawa.ca

2. Simpson’s paradox C. R. Charig et al. 1986. Br Med J (Clin Res Ed) 292 (6524): 879–882 Treatment A: all open procedures Treatment B: percutaneous nephrolithotomy Question: which treatment is better?

3. Applied Biostatistics • What is biostatistics: Biostatistics is for collecting, organizing, summarizing, presenting and analyzing data of biological importance, with the objective of drawing conclusions and facilitating decision-making. • Statistical estimation/description • point estimation (e.g., mean X = 3.4, slope = 0.37) • interval estimation (e.g., 0.5 < mean X < 8.5) • Significance tests • Statistic, e.g. t, F, 2 which are indices measuring the difference between the observed value and the expected value derived from the null hypothesis • Significance level and p value (e.g., p < 0.01) • Distribution of the statistic (We cannot obtain p value with the distribution)

4. Descriptive Statistics • Normal distribution: • Central tendency • dispersion • skewness • kurtosis. • Confidence Limits. -6 -4 -2 0 2 4 6 Confidence Limits: Mean ± t,dfSE

5. Example of data analysis Probability density function (PDF) Cumulative distribution function (CDF) Grouped data Null hypothesis Significance test and p value

6. Assignment 1 Compute mean, standard deviation (Std), standard error (SE), coefficient of variation (CV), and 95% confidence interval (lower and upper limits or LL and UL). you can use EXCEL function T.INV Group the data and do the same calculation for the grouped data Print this slide, fill in your answer, and hand at the beginning of class next Tuesday. Name: ID

7. Data: same pattern but smaller N Assignment 2: Do the same test of normality as in the slide with the original (large) N ( = 5738) and compare the two p values. No need to hand in, but get ready to discuss the result. The distribution of this data is nearly identical to the previous slide but N is smaller ( = 201)

8. Data: same pattern but smaller N Assignment 3: Do the same test of normality as the previous slide with comparable N. No need to hand in, but get ready to discuss the result

9. Decision making and risks 1. Type I error is also called Producer’s risk (rejecting a good product). To limit the chance of committing a Type I error, we typically set its rate () small.  is referred to as significance level in significance test. 2. Type II error is often referred to as consumer’s risk (accepting an inferior product), and its rate is typically represented by . One can avoid making Type II errors by making no decision until we have a sample size large enough to give us sufficient power to reject the null hypothesis. 3. The power of a test is 1- which depends on sample size and effect size.

10. Inference: Population and Sample Variable Variate ID Weight Length (in kg) (in m) 1 2.3 0.3 2 2.5 0.3 3 2.5 0.5 4 2.4 0.4 5 2.4 0.4 6 2.3 0.5 Mean 2.4 0.4 Population Sample Sample Individual Statistic Observation

11. Essential definitions • Statistic: any one of many computed or estimated statistical quantities, such as the mean, the standard variation, the correlation coefficient between two variables, the t statistic for two-sample t-test. • Parameter: a numerical descriptive measure (attribute) of a population. • Population: a specified set of individuals (or individual observations) about which inferences are to be made. • Sample: a subset of individuals (or individual observations), generally used to make inference about the population from which the sample is taken from.

12. Optional materials • Central tendency • Arithmetic mean • Geometric mean • Harmonic mean • Dispersion • Moments • Skewness • Kurtosis • Basic probability theory • Events and event space • Independent events • Mutually exclusive events • Joint events • Empirical probability • Conditional probability

13. Elementary Probability Theory • Empirical probability of an event is taken as the relative frequency of occurrence of the event when the number of observations is very large. • A coin is tossed 10000 times, and we observed head 5136 times. The empirical probability of observing a head when a coin is tossed is then5136/10000 = 0.5136. • A die is tossed 10000 times and we observed number 2 up 1703 times. What is the empirical probability of getting a 2 when the die is tossed? • If the coin and the die are even, what is the expected probabilities for getting a head or a number 2?

14. Mutually Exclusive Events • Two or more events are mutually exclusive if the occurrence of one of them exclude the occurrence of others. • Example: • observing a head and observing a tail in a single coin tossing experiment • events represented by null hypothesis and the alternative hypothesis • being a faithful husband and having extramarital affairs. • Binomial distribution

15. Coin-Tossing Expt. If I ask someone to toss a coin 6 times and record the number of heads, and he comes back to tell me that the number of heads is exactly 3. If I ask him repeat the tossing experiment three more times, and he always comes back to say that the number of heads in each experiment is exactly 3. What would you think? Experiment Outcome (Number of Heads out of 6 Coin-tossing)1 32 33 34 3The probability of getting 3 heads out of 6 coin-tossing is 0.3125 for a fair coin following the binomial distribution (0.5 + 0.5)6, and the probability of getting this result 4 times in a roll is 0.0095. The person might not have done the experiment at all!

16. Thinking Critically Now suppose Mendel obtained the following results: Breeding Experiment Number of Round Seeds Number of Wrinkled Seeds1 21 72 24 83 18 6 Based on (0.75+0.25)n: P1 = 0.171883; P2 = 0.161041; P3 = 0.185257; P = 0.0051 Edwards, A. W. F. 1986. Are Mendel’s results really too close? Biol. Rev. 61:295-312.

17. Compound Event • A compound event, denoted by E1E2 or E1E2…EN, refers to the event when two or more events occurring together. • For independent events, Pr{E1E2} = Pr{E1}Pr{E2} • For dependent events, Pr{E1E2} = Pr{E1}Pr{E2|E1}

18. Probability of joined events • Criteria Prob. • Between 25 and 45 1/2 • Very bright 1/25 • Liberal 1/3 • Relatively nonreligious 2/3 • Self-supporting 1/2 • No kids 1/3 • Funny, sense of humor 1/3 • Warm, considerate 1/2 • Sexually assertive 1/2 • Attractive 1/2 • Doesn’t drink or smoke 1/2 • Is not presently attached 1/2 • Would fall in love quickly 1/5 • The probability of meeting such a person satisfying all criteria is 1/648,000, i.e., • if you meet one new candidate per day, it will take you, on the average, 1775 years to find your partner. • Fortunately, many criteria are correlated, e.g., a very bright adult is almost always self-supporting.

19. Conditional Probability • Let E1 be the probability of observing number 2 when a die is tossed, and E2 be the probability of observing even numbers. The conditional probability, denoted by Pr{E1|E2} is called the conditional probability of E1 when E2 has occurred. • What is the expected value for the conditional probability of P{E1|E2} with a fair die? • What is the expected value for the conditional probability of P{E2|E1}?

20. Independent Events • Two events (E1 and E2) are independent if the occurrence or non-occurrence of E1 does not affect the probability of occurrence of E2, so that Pr{E2|E1} = Pr{E2}. • When one person throw a coin in Hong Kong, and another person throw a die in US, the event of observing a head and the event of getting a number 2 can be assumed to be independent. • The event of grading students unfairly and the event of students making an appeal can be assumed to be dependent.

21. Various Kinds of Means • Arithmetic mean • Geometric mean • Harmonic mean • Quadratic mean (or root mean square)

22. Geometric Mean • The geometric mean (Gx) is expressed as: • where  is called the product operator (and you know that  is called the summation operator.

23. When to Use Geometric Mean • The geometric mean is frequently used with rates of change over time, e.g., the rate of increase in population size, the rate of increase in wealth. • Suppose we have a population of 1000 mice in the 1st year (x1 = 1000), 2000 mice the 2nd year (x2 = 2000), 8000 mice the 3rd year (x3 = 8000), and 8000 mice the 4th year (x4 = 8000). This scenario is summarized in the following table: What is the mean rate of increase? (2+4+1) / 3 ?

24. Wrong Use of Arithmetic Mean • The arithmetic mean is (2+4+1) / 3 = 7/3, which might lead us to conclude that the population is increasing with an average rate of 7/3. • This is a wrong conclusion because1000 * 7/3 * 7/3 * 7/3  8000 • The arithmetic mean is not good for ratio variables.

25. Using Geometric Mean • The geometric mean is: • This is the correct average rate of increase. On average, the population size has doubled every year over the last three years, so that x4 = 1000 222 = 8000 mice. • Alternative: 1000*r3 = 8000

26. The Ratio Variable • Example: • Year 1: • Year 2: • The arithmetic mean ratio is r1 = 2.5 • What is the mean ratio of bread price to milk price? • Ratio r1 = 1/3; Ratio r2 = 1/2 • Arithmetic mean ratio is r2 = (1/3 + 1/2) / 2 = 5/12 = 0.4167. • But r1 1/r2. What’s wrong? • Conclusion: Arithmetic mean is no good for ratios

27. Using Geometric Mean • Geometric mean of the milk/bread ratios: • Geometric mean of the bread/milk ratios:

28. Moments and distribution • The moment (mr) • The central moment (r) • The first moment is the arithmetic mean • The second central moment • is the population variance when N is equal to population size (typically assumed to be infinitely large) • is the sample variance when N = n-1 where n is sample size • Standardized moment (r) = the moment of the standardized x. • 1 = 0 • 2 = 1 • 3 is population skewness; the sample skewness with sample size (n) is 2 = u2/n

29. Skewness Right-Skewed (+) Left-Skewed (-) -6 -4 -2 0 2 4 6

30. Kurtosis Leptokurtic(Kurtosis < 0) Normally distributed Platykurtic (Kurtosis > 0) -6 -4 -2 0 2 4 6

31. Chest Number of Men(inches)33 334 1835 81 36 185 37 420 38 74939 1073 40 1079 41 934 42 658 43 370 44 92 45 50 46 21 47 4 48 1 Marks Number of (mid-point) candidates 400 24 750 74 1250 38 1750 21 2250 11 2750 8 3250 11 3750 5 4250 2 4750 1 5250 3 5750 1 6250 0 6750 0 7250 0 7750 1 Empirical frequency distributions

32. SAS Program and Output Univariate Procedure Variable=CHEST N 5738 Sum Wgts 5738 Mean 39.83182 Sum 228555 Std Dev 2.049616 Variance 4.200925 Skewness 0.03333Kurtosis 0.06109 USS 9127863 CSS 24100.71 CV 5.145674 Std Mean 0.027058 T:Mean=0 1472.102 Pr>|T| 0.0001 Num ^= 0 5738 Num > 0 5738 M(Sign) 2869 Pr>=|M| 0.0001 Sgn Rank 8232596 Pr>=|S| 0.0001 D:Normal 0.098317 Pr>D <.01 USS = Sum(xi2) CSS = Sum(xi – MeanX)2 data chest; input chest number; cards; 33 3 34 18 35 81 36 185 37 420 38 749 39 1073 40 1079 41 934 42 658 43 370 44 92 45 50 46 21 47 4 48 1 ; proc univariate normal plot; freq number; var chest; run;

33. SAS Program and Output data Grade; input marks number; cards; 400 24 750 74 1250 38 1750 21 2250 11 2750 8 3250 11 3750 5 4250 2 4750 1 5250 3 5750 1 6250 0 6750 0 7250 0 7750 1 ; procunivariate normal plot; freq number; var marks; run; Univariate Procedure Variable= marks N 200 Sum Wgts 200 Mean 1465.5 Sum 293100 Std Dev 1179.392 Variance 1390965 Skewness 2.031081 Kurtosis 5.180086 USS 7.0634E8 CSS 2.768E8 CV 80.47708 Std Mean 83.39558 T:Mean=0 17.57287 Pr>|T| 0.0001 Num ^= 0 200 Num > 0 200 M(Sign) 100 Pr>=|M| 0.0001 Sgn Rank 10050 Pr>=|S| 0.0001 W:Normal 0.767621 Pr<W 0.0001

34. SAS Graph DATA; DO X=-5 TO 5 BY 0.25; DO Y=-5 TO 5 BY 0.25; DO Z=SIN(SQRT(X*X+Y*Y)); OUTPUT; END; END; END; PROC G3D; PLOT Y*X=Z/CAXIS=BLACK CTEXT=BLACK; TITLE 'Hat plot'; FOOTNOTE 'Fig. 1, Xia'; RUN;