Introduction to Statistical Data Analysis III

Transcription

1 Introduction to Statistical Data Analysis III JULY 2011 Afsaneh Yazdani

2 Preface Major branches of Statistics: - Descriptive Statistics - Inferential Statistics

3 Preface What is Inferential Statistics? The objective is to make inferences about a population parameters based on information contained in a sample.

4 Preface What is Inferential Statistics? The objective is to make inferences about a population parameters based on information contained in a sample. Mean, Median, Standard Deviation, Proportion

5 What is Inferential Statistics? Statistical inference-making procedures differ from ordinary procedures in that we not only make an inference but also provide a measure of how good that inference is.

6 What is Inferential Statistics? Methods for making inferences about parameters fall into two categories: - Estimating the population parameter - about a population parameter

7 Estimation Point Estimation The first step in statistical inference is Point Estimation, in which we compute a single value (statistic) from the sample data to estimate a population parameter.

8 Estimation Point Estimation Empirical Rule Central Limit Theorem Interval Estimation θ ± A SE(θ), where A is based on sampling distribution of θ

9 Estimation Estimation of : - Point Estimation: Sample Mean y

10 Estimation Estimation of : - Point Estimation: Sample Mean y - Interval Estimation:

11 Estimation Estimation of : - Point Estimation: Sample Mean y - Interval Estimation: For large n, y is approximately normally distributed with mean and standard error σ n

12 Estimation Estimation of : - Point Estimation: Sample Mean y - Interval Estimation: (y σ, y σ ) n n with level of confidence 95% when is known

13 Estimation Estimation of : - Point Estimation: Sample Mean y - Interval Estimation: (y σ, y σ ) n n with level of confidence 95% when is known In 95% of the times in repeated sampling, the interval contains the mean

14 Estimation Estimation of : - Point Estimation: Sample Mean y - Interval Estimation: (y s, y s ) n n with level of confidence 95% when is unknown

15 Estimation Estimation of : - Point Estimation: Sample Mean y - Interval Estimation: (y s n, y s n ) Is good approximation if population distribution is not too non-normal and sample size is large enough when is unknown

16 Estimation α % confidence Interval for ( known) when sampling from a normal population or n large ( y zα 2 s n, y + zα 2 s n )

17 Estimation α % confidence Interval for ( unknown) when sampling from a normal population or n large ( y tα 2 s n, y + tα 2 s n )

18 Estimation Goodness of inference for interval estimation: Confidence coefficient Width of the confidence interval

19 Estimation Goodness of inference for interval estimation: Confidence coefficient Width of the confidence interval Higher

20 Estimation Goodness of inference for interval estimation: Confidence coefficient Width of the confidence interval Smaller Higher

21 Estimation Sample Size Required for a α % Confidence Interval for of the Form y ± E n = zα 2 σ 2 2 E 2

22 Estimation Sample Size Required for a α % Confidence Interval for of the Form y ± E n = zα 2 σ 2 2 E 2 Estimate using information from prior survey

23 Estimation Sample Size Required for a α % Confidence Interval for of the Form y ± E n = zα 2 σ 2 2 E 2 Estimate using s= range 4

24 Estimation Sample Size for Estimation of : Consideration 2: Consideration 1: Desired Confidence Level (z-value) the tolerable error that establishes desired width of the interval Consideration 3: Standard Deviation ( ) ( y zα 2 n, y + zα 2 n ) appropriate sample size for estimating using a confidence interval

25 Estimation Sample Size for Estimation of : Consideration 1: High Confidence Level (z-value) Consideration 2: Narrow width of the interval Consideration 3: Large Standard Deviation ( ) ( y zα 2 n, y + zα 2 n ) Large Sample Size

26 Estimation Sample Size for Estimation of : Consideration 1: Confidence Level is traditionally set at 95% Consideration 2: Width of interval depends heavily on context of problem Consideration 3: Based on a guess about population standard deviation or Estimate from an initial sample ( y zα 2 n, y + zα 2 n ) Large Sample Size

27 Estimation Sample Size for Estimation of : desired accuracy of the sample statistic as an estimate of the population parameter required time and cost to achieve this degree of accuracy

28 Statistical Tests Using sampled data from the population, we are simply attempting to determine the value of the parameter. In hypothesis testing, there is a idea about the value of the population parameter.

29 Statistical Tests A statistical test is based on the concept of proof and composed of five parts: - H a : Research Hypothesis (Alternative Hypothesis) - H 0 : Null Hypothesis - Test Statistic - Rejection Region - Check assumptions and draw conclusions

30 Guidelines for Determining H 0 and H a in Statistical Tests - H 0 : the statement that parameter equals a specific value - H a : the statement that researcher is attempting to support or detect using the data - The null hypothesis is presumed correct unless there is strong evidence in the data that supports the research hypothesis.

31 Test Statistic The quantity computed from the sample data, that helps to decide whether or not the data support the research hypothesis.

32 Rejection Region The rejection region (based on the sampling distribution) contains the values of test statistic that support the research hypothesis and contradict the null hypothesis.

33 Statistical Tests - One-Tailed Test - Two-Tailed Test

34 Statistical Tests - One-Tailed Test The rejection region is located in only one tail of the sampling distribution of test statistic H a : θ < θ 0 Or H a : θ > θ 0

35 Statistical Tests - Two-Tailed Test The rejection region is located in both tails of the sampling distribution of test statistic H a : θ θ 0

36 Errors in Statistical Tests - Type I Error - Type II Error

37 Errors in Statistical Tests - Type I Error - Type II Error Rejecting the null hypothesis when it is true. The probability of a Type I error is denoted by.

38 Errors in Statistical Tests - Type I Error - Type II Error Accepting the null hypothesis when it is false. The probability of a Type II error is denoted by.

39 Errors in Statistical Tests Decision Reject Accept Null Hypothesis True Type I Correct (1- ) False Correct (1- ) Type II

40 Errors in Statistical Tests Decision Reject Accept Null Hypothesis True Type I Correct (1- ) False Correct (1- ) Type II The probabilities associated with Type I and Type II errors are inversely related. For a fixed sample size n, when decreases will increase and vice versa

41 Errors in Statistical Tests Decision Reject Accept Null Hypothesis True Type I Correct (1- ) False Correct (1- ) Type II Usually is specified to locate the Rejection Region

42 Errors in Statistical Tests Decision Reject Accept Null Hypothesis True Type I Correct (1- ) False Correct (1- ) Type II Power of Test

43 Effectiveness of a statistical test is measured by: Magnitudes of Type I Error and Type II Error

44 Effectiveness of a statistical test is measured by: For a fixed, as the sample size increases, decreases

45 Statistical Tests (Drawing Conclusion) Traditional Approach: - Using Statistic Test, two types of errors, their probability,

46 Statistical Tests Traditional Approach: - Using Statistic Test, two types of errors, their probability, The problem with this approach is that if other researchers want to apply the results of your study using a different value for then they must compute a new rejection region.

47 Statistical Tests Alternative Approach Using Level of Significance/P-Value Smallest size of at which H 0 can be rejected, based on the observed value of the test statistic.

48 Statistical Tests Alternative Approach Using Level of Significance/P-Value Smallest size of at which H 0 can be rejected, based on the observed value of the test statistic. The probability of observing a sample outcome more contradictory to H 0 than the observed sample result.

49 Statistical Tests Alternative Approach Using Level of Significance/P-Value The weight of evidence for rejecting the null hypothesis.

50 Statistical Tests Alternative Approach Using Level of Significance/P-Value The weight of evidence for rejecting the null hypothesis. The smaller the value of this probability, the heavier the weight of the sample evidence against H 0.

51 Statistical Tests Alternative Approach Decision Rule for Using P-Value P Value α Reject H 0 P Value > α Fail to Reject H 0

52 Statistical Test for population mean ( is known, when sampling from a normal population or n large) Test Statistic: z = y μ 0 σ n ~N(0, 1) H 0 : μ μ 0 H a : μ > μ 0 H 0 : μ μ 0 H a : μ < μ 0 H 0 : μ = μ 0 H a : μ μ 0 Reject H 0 if z z α Reject H 0 if z z α Reject H0if z zα 2

53 Statistical Test for population mean ( is known, when sampling from a normal population or n large) Power of the test: One-Tailed Test 1 β μ a = 1 Pr(z z α μ 0 μ a σ n ) Two-Tailed Test 1 β μ a 1 Pr(z zα 2 μ 0 μ a σ n )

54 Statistical Test for population mean ( is known, when sampling from a normal population or n large) H 0 : μ μ 0 H a : μ > μ 0 P-Value: Pr (z computed z) H 0 : μ μ 0 H a : μ < μ 0 P-Value: Pr (z computed z) H 0 : μ = μ 0 H a : μ μ 0 P-Value: 2Pr (z computed z )

55 Statistical Test for population mean ( is unknown, when sampling from a normal population or n large) Test Statistic: T = y μ 0 s n ~t(n 1) H 0 : μ μ 0 H a : μ > μ 0 H 0 : μ μ 0 H a : μ < μ 0 H 0 : μ = μ 0 H a : μ μ 0 Reject H 0 if t t Reject H 0 if t t α Reject H0if t tα 2

56 Statistical Test for population mean ( is unknown, when sampling from a normal population or n large) H 0 : μ μ 0 H a : μ > μ 0 P-Value: Pr (t computed t) H 0 : μ μ 0 H a : μ < μ 0 P-Value: Pr (t computed t) H 0 : μ = μ 0 H a : μ μ 0 P-Value: 2Pr (t computed t )

57 Non-normal distribution of population 2 ISSUES to consider: - Skewed Distribution - Heavy-tailed Distribution

58 Non-normal distribution of population 2 ISSUES to consider: - Skewed Distribution - Heavy-tailed Distribution Tests of Hypothesis tend to have smaller than specified level, so test has lower power

59 Non-normal distribution of population 2 ISSUES to consider: - Skewed Distribution - Heavy-tailed Distribution Tests of Hypothesis tend to have smaller than specified level, so test has lower power Robust Methods

60 Inferences about Median When the population distribution is highly skewed or very heavily tailed or sample size is small, median is more appropriate than the mean as a representation of the center of the population.

61 Statistical Test for population median (Sign Test) Test Statistic: W i = y i M 0, B = No. of Positive W i s B~Binom(n, π) H 0 : M M 0 H a : M > M 0 H 0 : M M 0 H a : M < M 0 H 0 : M = M 0 H a : M M 0 Reject H 0 if B n C 1,n Reject H 0 if B C α 1,n Reject H0if B Cα 2,n or B n C α 2,n

62 Statistical Test for population median (Approximation) Test Statistic: B st = B n 2 n 4, B st ~N(0, 1) H 0 : M M 0 H a : M > M 0 H 0 : M M 0 H a : M < M 0 H 0 : M = M 0 H a : M M 0 Reject H 0 if B st z with P-value Pr(z B st ) Reject H 0 if B st z with P-value Pr(z B st ) Reject H0if Bst zα 2 2Pr(z B st ) with P-value

63 Statistical Test for Mean (one-population) Mean of One- Population Normal Population, or n large σ known Test using z Mean of One- Population Normal Population, or n large σ unknown T-Test Mean of One- Population Non-normal Population Sign-Test

64 Statistical Test for population Variance (Normal Population) Test Statistic: H 0 : σ 2 2 σ 0 H a : σ 2 2 > σ 0 H 0 : σ 2 2 σ 0 H a : σ 2 2 < σ 0 H 0 : σ 2 2 = σ 0 H a : σ 2 2 σ 0 2 = (n 1)s2 σ 0 2 ~ 2 (n 1), s 2 = Reject H 0 if 2 2 > U,α Reject H 0 if 2 < L, 2 Reject H 0 if 2 2 > α U, 2 or 2 2 > α L, 2 n i=1 n 1 y i y 2

65 100 1 α % confidence Interval for σ 2 (or σ) (n 1)s 2 U 2 < σ 2 < (n 1)s 2 < σ < 2 U (n 1)s2 L 2 (n 1)s2 L 2

66 The inferences we have made so far have concerned a parameter of a single population. Quite often we are faced with an inference involving a comparison of parameters of different populations

67 Theorem If two independent random variables y 1 and y 2 are normally distributed with means and variances (μ 1, σ 1 2 ) and (μ 2, σ 2 2 ) respectively, then (y 1 y 2 )~N μ 1 μ 2, σ σ 2 2

68 Sampling Distribution for y 1 y 2 Two independent large samples (y 1 y 2 )~ N μ 1 μ 2, σ n 1 + σ 2 n 2

69 Statistical Test for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 = σ 2 2 ) Test Statistic: t= (y 1 y 2 ) D 0 s p where: n 1 n 2 ~t(n 1 + n 2 2) s p = n 1 1 s n 2 1 s 2 2 n 1 +n 2 2

70 Statistical Test for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 = σ 2 2 ) Test Statistic: t= (y 1 y 2 ) D 0 s p where: n 1 n 2 ~t(n 1 + n 2 2) D 0 is a specified value, often 0 s p = n 1 1 s n 2 1 s 2 2 n 1 +n 2 2

71 Statistical Test for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 = σ 2 2 ) Test Statistic: s p is a weighted average, that combine (pools) two independent estimates of σ t= (y 1 y 2 ) D 0 s p where: s p = n 1 n 2 ~t(n 1 + n 2 2) n 1 1 s n 2 1 s 2 2 n 1 +n 2 2

72 Statistical Test for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 = σ 2 2 ) Test Statistic: t= (y 1 y 2 ) D 0 s p n 1 n 2 ~t(n 1 + n 2 2) H 0 : μ 1 μ 2 D 0 H a : μ 1 μ 2 > D 0 H 0 : μ 1 μ 2 D 0 H a : μ 1 μ 2 < D 0 H 0 : μ 1 μ 2 = D 0 H a : μ 1 μ 2 D 0 Reject H 0 if t t Reject H 0 if t t α Reject H0if t tα 2

73 100 1 α % confidence Interval for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 = σ 2 2 ) (y 1 y 2 ) ± tα 2 s p 1 n n 2

74 Cluster-Effect Serial or Spatial Correlation, Paired Sample Dependency in observations More Advanced methods such as longitudinal or spatial analysis, Paired Tests Skewness Heavy-tailed Non- Normality Non-Parametric Tests Un-equal variances Approximate T-test

75 Statistical Test for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 σ 2 2 ) Test Statistic: where: t = (y 1 y 2 ) D 0 s s 2 2 n 1 n 2 ~ t(df) df = (n 1 1)(n 2 1) and c = s 1 1 c 2 n 1 1 +c 2 (n 2 1) s n1 + s 2 2 n 1 n 2

76 Statistical Test for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 σ 2 2 ) Test Statistic: t ~ t(df) H 0 : μ 1 μ 2 D 0 H a : μ 1 μ 2 > D 0 H 0 : μ 1 μ 2 D 0 H a : μ 1 μ 2 < D 0 H 0 : μ 1 μ 2 = D 0 H a : μ 1 μ 2 D 0 Reject H 0 if t t Reject H 0 if t t α Reject H0 if t tα 2

77 100 1 α % confidence Interval for μ 1 μ 2 (Independent samples, y 1 and y 2 approximately normal, σ 1 2 σ 2 2 ) 2 s (y 1 y 2 ) ± 1 tα + s n 1 n 2

78 Statistical Test for μ 1 μ 2 Independent Samples, Wilcoxon Rank Sum Test 1- Sort the data and replace the data value with its rank 2- Make the Test Statistic: - when n 1, n 2 10 then T=sum of the ranks in sample 1 - when n 1, n 2 > 10 then z = T μ T σ T μ T = n 1(n 1 +n 2 +1), and σ 2 T = n 1n 2 (n n 2 + 1)

79 Statistical Test for μ 1 μ 2 Independent Samples, Wilcoxon Rank Sum Test 1- Sort the data and replace the data value with its rank 2- Make the Test Statistic: - when n 1, n 2 10 then T=sum of the ranks in sample 1 - when n 1, n 2 > 10 then z = T μ T σ T Normal Approximation μ T = n 1(n 1 +n 2 +1), and σ 2 T = n 1n 2 (n n 2 + 1)

80 Statistical Test for μ 1 μ 2 Independent Samples, Wilcoxon Rank Sum Test 1- Sort the data and replace the data value with its rank 2- Make the Test Statistic: - when n 1, n 2 10 then T=sum of the ranks in sample 1 - when n 1, n 2 > 10 then z = T μ T σ T μ T = n 1(n 1 +n 2 +1), and σ 2 T = n 1n 2 (n n 2 + 1) Provided there are no tied ranks

81 Statistical Test for μ 1 μ 2 Independent Samples, Wilcoxon Rank Sum Test Test Statistic: z = T μ T σ T H 0 : Two populations are identical H a : Population 1 is shifted to the right of population 2 H a : Population 1 is shifted to the left of population 2 H a : Population 1 and 2 are shifted from each other Reject H 0 if z z α Reject H 0 if z z α Reject H 0 if z zα 2

82 Statistical Test for μ d (Paired samples, y 1 y 2 approximately normal) Test Statistic: t = d D 0 s d n ~t(n 1) H 0 : μ d D 0 H a : μ d > D 0 H 0 : μ d D 0 H a : μ d < D 0 H 0 : μ d = D 0 H a : μ d D 0 Reject H 0 if t t Reject H 0 if t t α Reject H0 if t tα 2

83 100 1 α % confidence Interval for μ d (Paired samples, y 1 y 2 approximately normal) d ± tα 2 s d n

84 Statistical Test for μ 1 μ 2 Paired Samples, Wilcoxon Signed-Rank Test 1- Calculate differences of the pairs, subtract them from D 0 keep non-zero differences (n), sort the absolute values in increasing order and rank them. 2- Make the Test Statistic: - when n 50 then T, T +, or min(t, T + ) depending on H a - when n > 50 then z = T n(n+1) 4 n(n+1)(2n+1) 24

85 Statistical Test for μ 1 μ 2 Paired Samples, Wilcoxon Signed-Rank Test H 0 : M = D 0 H a : M > D 0 Reject H 0 if z < z α H 0 : M = D 0 H a : M < D 0 Reject H 0 if z < z α H 0 : M = D 0 H a : M D 0 Reject H 0 if z < zα 2

86 Statistical Test for Mean (Two-population) Means of Two- Populations Independent Samples Normal Population, or n large σ 1 = σ 2 T-Test (Equal Variances) Means of Two- Populations Independent Samples Normal Population or n large σ 1 σ 2 T-Test (Unequal Variances) Means of Two- Populations Independent Samples Non-normal Population Wicoxon Rank Sum Test

87 Statistical Test for Mean (Two-population) Means of Two- Populations Paired Samples Normal Population, or n large Paired T-Test Means of Two- Populations Paired Samples Non-normal Population Wicoxon Signed Rank Test

88 Statistical Test for σ 1 2 σ 2 2 (Normal Population) Test Statistic: F = s 1 2 s 2 2 ~F(n 1 1, n 2 1) H 0 : σ σ 2 H a : σ > σ 2 H 0 : σ = σ 2 H a : σ σ 2 Reject H 0 if F F α,df1,df 2 Reject H 0 if F F 1 α 2,df 1,df 2 or if F Fα 2,df 1,df 2

89 100 1 α % confidence Interval for σ 1 2 σ 2 2 s 1 2 s Fα 2,df 1,df 2 < σ σ 2 2 < s 1 s 1 2 Fα 2,df 2,df 1