Stats 256 Exam
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| Concordant | A pair is concordant if
(xi-xj)*(yi-yj) > 0
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| Discordant | A pair is discordant if
(xi-xj)*(yi-yj) < 0
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| Non-informative Censoring | We assume that censored individuals are at the same risk of failure as those who are still alive and uncensored; This implies that the censoring process is independent of survival time
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| Turnbull estimator | Used for interval censored data, assumes non-informative censoring
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| Interval censored data | When the time of death is known to happen between two points, but the exact time is unknown. The nonparametric survival function for this type of data is not continuous. Left and right censored are types of this
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| Survival Function | Non-increasing function, value 1 at time 0 because it deals with probabilities, 0 at time = infinity. Same as CDF but backwards (1-CDF) = survival function, CDF = 1-S(f)
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| Right Censored Data | A subjects actual time-to-event is known only to occur after a given time (they dropped out, study ended)
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| Left-censored Data | Arises when Ti is known only to occur before some censoring time
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| Kaplan Meier Estimator | The limit of the life-table estimator when intervals are taken so small that only at most one distinct observation occurs within an interval; used only for right censored or exact data; step function; jumps down at death
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| Signed Rank Test (wilcoxon Signed rank test) | Assume no ties and no diff=0
Calculate Paired Differences & abs value of paired differences
Rank absolute value
Attach original sings
Compute SR+
(Can be used for paired data)
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| SR+ for a signed rank test | Observed value of the sum of the positive signed ranks
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| P-value for a Signed Ranked Test | Fraction of SR+ values that are greater than or equal to SR+observed (one-sided) over 2^n (sample size)
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| Ties and 0's in Signed Rank Test | Average ties as usual
Use a method of ranking with zeros or omit zeros (SAS omits zeros)
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| Zeros in the Signed Rank Test | If there are not too many, ranking with or without them will give similar results
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| Sign Test | (can be used for paired data, does not consider ranks)
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| SN+ in the sign test | The number of positive differences; follows the binomial distribution with p=.5
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| Sign Test Hypotheses | H0: Ɵd = 0 or Ha: Ɵd≠ 0
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| Exact P-value for Sign Test | 1 - binomcdf(n, probofsuccess, value of interest)
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| Large Sample Approximation for Sign Test | (SN+ - n/2) / ( sqrt(n/4)) = Z
Then use normcdf to get p-value
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| Advantages of the Sign Test | Easy to perform
Protects against outliers
Efficient for heavy tailed distributions
Don't need actual data, just signs of differences
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| Disadvantages of Sign Test | Not as powerful as Signed Rank Test
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| LogRank Test Assumption | The times of interest are either exactly known or right-censored
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| Logrank Test Hypotheses | H0: F0(t) = F1(t) <-- CDF
...Since S(t) = 1-F(t)
S0(t)= S1(t)
Ha: Survival functions are not equal (two-sided)
Could also be not equal w/ strict inequality for at least one time, t
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| Pearson Correlation Coefficient | Measures strength, direction of LINEAR data
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| Hypotheses for Pearson Correlation Coefficient | rho equals 0, rho does not equal 0
correlation, no correlation
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| Estimated Slope and Pearson's CC | r * (Sy / Sx)
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| Assumptions for Pearson's CC | Quantitative/ Numerical Data
Linearity
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| Permutation Test for Rho | Same hypotheses
n! possible permutations
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| Steps for Permutation test for Rho | Calculate robs (observed correlation)
Permute y's among x's n! ways
Calculate r for each permutation
Get p-value
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| P-value for Permutation test for Rho | P-value =
# of r values >= robs / (R <--permutations we consider)
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| Spearman Correlation Coefficient | Measures extent to which y increases with x by comparing the ranks of the x's (1 to n) with the ranks of the y's (1 to n)
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| Benefits of Spearman's Correlation Coefficient | Doesn't need quantitative data, only need to be able to rank it
Doesn't need to look linear
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| Kendall's Tau | A measure of association between two variables based on counts of concordant and discordant pairs
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| Calculation Tau | 1 point for Concordant Pairs
0 Points for discordant Pairs
.5 points for tied pairs
Total points and
Tau = 2((totalpoints)/(nchoose2)) - 1
Approximately Distributed Normally
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| Chi Square Test | Individuals are placed in two categories based on two overlapping characteristics
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| Hypotheses for Chi Square Test (SRS) | "Testing independence between rows and columns"
H0 : Pij = Pi.*P.j
Ha : Not H0 (not independent)
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| Hypotheses for Chi Square Test (Stratified of Completely Randomized Design) | "Test of Homogeneity"
H0 : Pi|j = Pi|j' (no association between rows and columns)
Ha : Not H0
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| Kruskal Wallis Test | Used to obtain a nonparametric rank test for comparing K treatments. Test Statistic is equivalent to F stat (applied to ranks). Natural extension of WRST for location(center of distribution)
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| Hypotheses for Kruskal Wallis Test | H0: F1(x) = F2(x) =...=...=Fn(x)
Ha: At least one F(x) is different
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| Permutation Distribution based on Kruskal Wallis Test stat | Has Chi-squared distribution with k-1 DF.
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| Pairwise Comparisons | # of Treatment Groups choose 2 = possible comparisons
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| Bonferroni | Alpha Adjustment Technique...
Alpha / (# of comparisons)
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| Bonferroni in the NonParametric Setting | Sample mean Ranks are used instead of sample means
N(N+1)/12 * sample variance instead of sqrtMSE
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| Kruskal Wallis Test Steps | 1. Rank Data across all trt groups
2. Find average of ranks btwn groups
Get K statistic
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| Possible Permutations in Kruskal Wallis | (Number of Obs!) / (n1!n2!...nk!)
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| p-value for K-W Stat | Calculate p-value stat for all possible permutation=KW*; # of KW*>= Observed / total permutations considered
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| Experiment-Wise Error Rate | Committing a type 1 error in multiple comparison tests is greater than in 2 sample test
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| Bonferroni | Alpha adjustment technique (new alpha is FWalpha/number of comparisons); can be too conservative
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| Protected LSD | Only run multiple comparisons is F-test is significant
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| Tukey's HSD | based on Q distribution; effect significant if mean differences are greater than qstat*sqrt(MSE/n)
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| Tukey | Invented Box-plot, stem and leaf plot, HSD & q statistic
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| Pooled Standard Deviation (Sp) | sqrt(MSE) ..(from ANOVA table)
Use all n points in formula... (n1-1)S1^2/(n1) etc
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| Standard Error | Sp* sqrt(1/n1 + 1/n2)
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| Fcrit for multiple comparisons | Use DF = N-K
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| Bonferroni Confidence Interval | width for each pairwise comparison is always the same
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| Reasons to adjust for multiple comparisons | Overall confidence interval goes down
Probability of family wise type one error would go up
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| Kolmogrov Smirnov Test | Designed to detect differences in location (center), scale (variability), or shape of two distributions
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| Two sample t-test has correct type1 error rate and highest power among unbiased tests if... | the populations are normal with known but equal variances
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| Conclusion | A statement about the alternative hypothesis
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| Permutation test | Any test that finds the p-value as the proportion of regroupings that lead to a statistic as extreme or more extreme than what was observed
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| Test the median when | The only assumption met is SRS (normality is violated, small sample size, skewness and outliers)
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| Power | probability of correctly rejecting the null; (1-Beta)
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| Bernoulli Trial | a trial or experiment with two possible outcomes
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| Ansari-Bradley | Test on variances, won't work if medians are different, Rank from both ends, C=Sum of group 1 ranks.
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| Omnibus Test | A test designed to pick up differences among treatments regardless of the nature of the differences between them
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| Correction for Ansari-Bradley | If medians are different, make then equal w/ addition or subtraction, apply to entire data set and re-run test
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| Permutation Principle | Says that the permutation distribution is an appropriate reference distribution for determining the p-value for a test
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| The type 1 error for a T-test will be close to alpha for large samples from any continuous distribution because of... | the central limit theorem
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| Permutation | A rearrangement of objects in which the order does matter
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| Disadvantages of a permutation test | Can be time consuming, large sample sizes require a lot of regroupings
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| T-test vs. wilcoxon | Heavy tailed distribution = wilcoxon;
light tailed = t-test
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| A symmetric distribution | the binomial distribution when Pi=.5
(left >.5; right <.5)
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| Combination | A rearranging of objects in which order does NOT matter
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| U | The number of pairs for which Xi > Yj
(if x=y we add .5 to U)
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| Efficiency of a test A to B | eff(AtoB) = NB/NA ... if eff >1 then A requires a smaller sample size; if eff <1 then B does
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| Mann Whitney & Wilcoxon Rank Sum | Are equivalent in that they are a function of one another (W = ((N(N+1))/2) + U)
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| K-S test statistic | The maximum W stat times the absolute value of the difference between the two estimated W stats
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| n! / (n-r)! | The number of ways to choose r things from a total of n things (nPr)
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| Characteristics of a binomial Experiment | 1.there are n bernoulli trials where n is known in advance
2. X is the number of successes in n trials
3. n trials are independent
4. true probability (Pi) is the same for every trial
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| Nonparametric Methods | 1.Binomial Distribution
2. Permutation
3. Bootstrap resampling
4. Smoothing and Non-least squares
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| (n choose r) | n!/(r!(n-r)! if order doesnt matter....nCr
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| Independence | 2 things are independent if the probability of one event occurring does not effect the probability or another event occurring
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| Theta.5 vs ThetaH | True Median vs Some hypothesized value
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| Advantages of a Permutation test | Used on small sample sizes, no normality assumption, no equal variances
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| tcdf function | tcdf(Tobt, 9999, DF) [multiply by two if you need two-sided]
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| binomCDF function | binomCDF(n,probability,value of interest) : does probability less than & equal to value of interest (times 2 for two sided)
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| CDF graph | Probability on Y axis, littlex value on x axis, step function, open points on right side
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| Exact Test on Binomial Distribution | WITH A CALCULATOR
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| Approximate Method for the Binomial Distribution | Z = B-.5n / sqrt(.25n) :
P(Z>=Zobt) <- plug into NormCDF
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| NormCDF function | (Zobt, 9999, mean,stddev)
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| Continuity Correction | B- or B+ .5 approximate method formula
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| advantages to the sign test | only looked at positive or negative signs (paired test so indicates type of difference btwn pairs)
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| Disadvantages to the sign test | Didn't account for magnitude of the differences
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