Assignment #6. Chapter 10: 14, 15 Chapter 11: 14, 18. Due tomorrow Nov. 6 th by 2pm in your TA s homework box

Transcription

1 Assignment #6 Chapter 10: 14, 15 Chapter 11: 14, 18 Due tomorrow Nov. 6 th by 2pm in your TA s homework box

2 Assignment #7 Chapter 12: 18, 24 Chapter 13: 28 Due next Friday Nov. 13 th by 2pm in your TA s homework box

3 Reading For Today: Chapter 14 For Tuesday: Chapter 15

4 Lab Report Posted on web-site Dates Rough draft due to TAs homework box on Monday Nov. 16 th Rough draft returned in your lab section the week of Nov. 23 rd Final draft due at start of your registered lab section the week of Nov. 30 th 10% of course grade Rough Draft - 5% Final draft - 5% If you re happy with your rough draft mark, you can tell your TA to use it for the final draft Read the Writing a Lab Report section of your lab notebook for guidance!!

5 Chapter 13 Review

6 Assumptions of t-tests Random sample(s) Populations are normally distributed (for 2-sample t) Populations have equal variances

7 Detecting deviations from normality Previous data/ theory Histograms Quantile plots Shapiro-Wilk test

8 Sampled from a normally distributed population

9 Sampled from non-normally distributed populations

10 Detecting deviations from normality: by quantile plot Normal data

11 Detecting differences from normality: Shapiro-Wilk test A Shapiro-Wilk test is used to test statistically whether a set of data comes from a normal distribution.

12 What to do when the assumptions are not true If the sample sizes are large, sometimes the parametric tests work OK anyway Transformations Non-parametric tests Randomization and resampling

13 Data transformations A data transformation changes each data point by some simple mathematical formula.

14 Log-transformation [ ] Y " = ln Y Frequency Y Y' = ln[y]

15 Other transformations Arcsine p " = arcsin[ p] proportions Square-root Y " = Y +1 2 Counts; When standard deviaiton and mean increase Square " Reciprocal Y = Y 2 " Y = 1 Y together Left skwed data Right skewed data Antilog Y " = e Y Left skewed data

16 Non-parametric methods Assume less about the underlying distributions Also called "distribution-free" "Parametric" methods assume a distribution or a parameter

17 Sign test Non-parametric test Compares data from one sample to a constant Simple: for each data point, record whether individual is above (+) or below (-) the hypothesized constant. Use a binomial test to compare result to 1/2.

18 The sign test has very low power So it is quite likely to not reject a false null hypothesis.

19 Most non-parametric methods use RANKS Rank each data point in all samples from lowest to highest Lowest data point gets rank 1, next lowest gets rank 2,...

20 Non-parametric test to compare 2 groups The Mann-Whitney U test compares the central tendencies of two groups using ranks.

21 Performing a Mann-Whitney U test First, rank all individuals from both groups together in order (for example, smallest to largest) Sum the ranks for all individuals in each group --> R 1 and R 2

22 Calculating the test statistic, U U = n n + n ( n +1) R 1 U 2 = n 1 n 2 U 1 U 1 is the number of times an individual from pop. 1 has a lower rank than an individual from pop. 2, out of all pairwise comparisons.

23 Mann-Whitney: Large sample approximation For n 1 and n 2 both greater than 10, use Z = 2U n 1 n 2 n n ( n + n +1) / Compare this Z to the standard normal distribution

24 Permutation tests Also known as randomization tests Used for hypothesis testing on measures of association Mixes the real data randomly Variable 1 from an individual is paired with variable 2 data from a randomly chosen individual. This is done for all individuals. The estimate is made on the randomized data. The whole process is repeated numerous times. The distribution of the randomized estimates is the null distribution.

25 Real data: Male wingless Y 1 Y 2 = 1.41 Male winged Randomized data: Male wingless Male winged Y 1 Y 2 =

26 1000 permutations P < 0.001

27 Chapter 14 Designing Experiments

28 Types of studies Experimental study Researchers assign treatments to units so that differences in response can be compared. Observational Study Researcher has no influence over which subjects receive which treatments.

29 Why do experimental study? Random assignment of treatments minimizes influence of confounding variables Confounding variables mask or distort the causal relationship between measured variables in a study

30 Confounding variables Unmeasured variable that masks or distorts the causal relationship between measured variables in a study Supplemental Oxygen (Explanatory variable) Survive Mt. Everest (Response variable) Preparedness (Confounding variable)

31 Goals of experiments Eliminate bias Reduce sampling error (increase precision and power)

32 Precise Imprecise Unbiased Biased

33 Design features that reduce Controls bias Random assignment to treatments Blinding

34 Controls A group which is identical to the experimental treatment in all respects aside from the treatment itself.

35 Uncontrolled experiment Treatment applied to group of subjects and response measured. We cannot determine whether the treatment is the cause of the response.

36 Example: placebo Some illnesses, e.g. pain and depression, respond to fact of treatment, even with no pharmaceutically active ingredients Control: "sugar pills"

37 Example: independent recovery Patients tend to seek treatment when they feel very bad As a result, they often visit the doctor when they are at their worst. Improvement may be inevitable, even without treatment Control: untreated group to compare with, if we want to measure the effects of a new therapy

38 Example: Stress associated with experimental methods Stressful or intrusive methods may produce a response separate from the effect of the treatment of interest Control: use same methods on group that does not get treatment of interest

39 Randomization The random assignment of treatments to units in an experimental study Breaks the association between possible confounding variables and the explanatory variable.

40 Randomization Supplemental Oxygen (Explanatory variable)? Survive Mt. Everest (Response variable) Preparedness (Confounding variable)

41 Randomization Doesn t eliminate variation caused by confounding variable, only their correlation with treatment Variation from confounding variables is spread more evenly between treatments, so they create no bias.

42 Randomize using a random process Example: Random number generator on computer (e.g. random.org) 1. List all subjects 2. Assign each a random number 3. Assign treatment A to lowest numbers and B to highest numbers.

43 Experiment: individuals are randomly assigned to treatments

44 Examples of wrong ways to randomize Treatment A to all patients at one clinic and B to all patients at second clinic Assign treatments alphabetically Haphazard assignment (researcher trying to be random)

45 Blinding Preventing knowledge of patient and/or experimenter of which treatment is given to whom Single blind blind patient Double blind blind patient and experimenter Unblinded studies usually find much larger effects (sometimes threefold higher), showing the bias that results from lack of blinding

46 Reducing sampling error Increasing the signal to noise ratio t = Y 1 Y 2 # 1 s & p % ( $ n n ' 1 2 "Signal" "Noise"

47 Reducing sampling error Increasing the signal to noise ratio 2" 1 s p $ + 1 If the "noise" # n 1 n 2 & is smaller, it is easier to detect a given "signal". % '. Can be achieved with smaller s or larger n.

48 Design features that reduce the effects of sampling error Replication Balance Blocking Extreme treatments

49 Replication The application of every treatment to multiple, independent experimental units

50 Replication

51 Replication SE Y1 Y2 " 1 = s 2 p $ + 1 # n n 1 2 % ' & Larger n reduces sampling error

52 What are experimental units? Units that are randomly sampled and assigned treatments Single individuals Batches of individuals that are more similar to each other than to other batches (e.g. family) Pseudoreplication (using more experimental units than you actually have) causes underestimation of standard errors and P-values

53 Balance In a balanced experimental design, all treatments have equal sample size.

54 Balance increases precision # 1 SE = s Y 1 Y p % 2 $ n n 1 2 & (. ' For a given total sample size (n 1 +n 2 ), the standard error is smallest when n 1 =n 2.

55 Balance increases precision n 1 +n 2 =20 n =10 1 n = = 0.2 n n 1 2 n 1 =19 n 2 =1 1 n n 2 =1.05

56 Blocking The grouping of experimental units that have similar properties. Within each block, treatments are randomly assigned to experimental units.

57 Blocking accounts for extraneous variation C = Control T = Treated Variance among hospitals will not contribute to SE. Only variance within hospitals will contribute to "noise"

58 Paired design is an example of blocking Treatment effects are measured by differences between treatments within pairs. This minimizes the influence of differences between pairs.

59 Randomized block design Like a paired design but for more than two treatments.

60 Extreme Treatments Treatment effects are easiest to detect when they are large. Stronger treatments can increase the signal-to-noise ratio. Caution: effects may not scale linearly

61 Experiments with more than one factor A factor is a single treatment variable whose effects are of interest to the researcher Multiple factors to: Make more efficient use of money and resources Estimate effects of interaction between factors

62 Interaction between explanatory variables The effect of one variable depends on the state of a second variable

63 Factorial Design Investigates all treatment combinations of two or more variables. Can measure interactions between treatments

64 Example of factorial design and interaction

65 What if we can t do experimental studies? Best observational studies Observational studies are still useful to detect patterns and generate hypotheses Minimize bias: Controls Randomization Blinding Minimize sampling error: Replication Balance Blocking Extreme treatments

66 Matching Every individual in the treatment group is paired with a control individual having the same or very similar values for the suspected confounding variables Does not account for all confounding variables (like randomization does), but only those used to match participants.

67 In-class Exercise Do people use more paper when they know it will be recycled? People given paper and told to test scissors. Recycling bin wither present or not No recycling bin: 4,4,4,4,4,4,4,5,8,9,9,9,9,12,12,13,14,14,14,14,15,23 Recycling bin: 4,5,8,8,8,9,9,9,12,14,14,15,16,19,23,28,40,43,129, Make histograms and identify options for test 2. Choose an test that you can do in class and conduct it