Quantitative Methods in Fashion Psychology from London College of Fashion

Quantitative Methods in Fashion Psychology

Week 19 Lecture: Hypothesis Testing

Sample Parametric vs. Non- Parametric tests Effect Size Power

Lecture Presenters and Developers

• Dr Jekaterina Rogaten
• j.rogaten@fashion.arts.ac.uk

The Lecture is developed by:

• Dr Jekaterina RogatenOverview of the session
• Hypotheses testing
• Type I and Type II error
• Inferential statistics
• Statistical significance
• One- and two-tailed tests
• Research Sample
• Parametric vs Non-parametric tests
• Power and practical significance

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What is Hypothesis

Clearly stated hypotheses direct attention to particular aspects of the research and help to choose the design of the study and measures Perhaps the main use of statistical techniques is to test hypotheses Hypotheses: Short clear statement of an educated guess about how things work. Reflecting your research question. Should be testable using statistical test.

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Alternative Hypothesis & Null Hypothesis

The null hypothesis H0 : Ho: 11 = 12 (The DV will not differ between the treatment and control conditions) It is a statistical statement and only can be expressed through formula The non-directional alternative/research hypothesis H : H1: 11 # 12 (The DV will differ between the treatment and control conditions) The directional alternative/research hypotheses H : H1: 11 > 12The DV will be higher in the test than in the control condition H1: 11 < 12The DV will be higher in the control than in the test condition

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Statistical Hypothesis Testing

A process by which decisions are made concerning the value of parameters Assume that we conduct an experiment and find that the mean scores of the DV across levels of the IV are consistent with the research hypothesis . Why do we need inferential statistics? Because the results could be due to chance Thus, only if we can disconfirm the null hypothesis do we claim that our data represent a real relationship/difference This can be achieved by determining the probability of obtaining a particular sample relationship/difference by chance alone

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Statistical Testing and Null Hypothesis

Statistical testing is based on the probability p(data/H0 ) that the observed data are consistent with the null hypothesis H0 The presumption is that any observed difference between conditions is due to chance alone If p(data/H0 ) is small, then H0 is rejected and we conclude that the research (alternative) hypothesis is supported by the findings Otherwise, H0 is retained and we conclude that the research (alternative) hypothesis is not supported by the findings NOTE: within the research report, you should only refer to the research hypotheses. NEVER MENTION NULL HYPOTHESIS ONE DOES NOT SIMPLY SUPPORT A NULL HYPOTHESIS @luminousmen.com

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Statistical Tests for Hypotheses

Comparison of two groups Comparison of three or more groups Measure association between two vaariables Prediction Description of one group Comparison of one group to a hypothetical value

Unpaired Groups

Unmatched Groups

Paired Groups

Matche From another measured variable From several measured or binomial variables 0 N R. I O N R. I Chi-square or Binomial test ND NN D Fisher's test (chi-square for large samples) R. I o N R,I O N R,I N Cochrane 0 NND ND NND ND Multiple logistic regressions Repeated measure S ANOVA R.I O N R, I O N 0 N McNemar's test ND NND Contingency coefficients Simple logistic regression Proportion N NND N NND Paired t R. I o N Wilcoxon test Spearman correlation ND NND Chi-square test Multiple linear regressions or Multiple nonlinear One-way ANOVA Kruskal-Wallistest Pearson correlation R, I = Ratio and Interval data O= Ordinal data N = Nominal data N = Normal distribution NND = Non normal distribution

ual: STATISTICS TO TEST HYPOTHESES AND YOUR PATIENCE NOS ND NND One- sample t test Wilcoxon test Unpaired t test Mann-Whitney test Friedman test Simple linear regression Nonparam etric regression R, I Mean, SD Median, interquartile range 7

Statistical Hypothesis Testing: Significance Level

• Formally, Ho is rejected if p(data/H0) < a
• a is called the significance level
• How small should a be?
• There is no right answer
• The common choice is a = . 05 . But there is nothing magic about .05! In p<0.05 We trust

Statistical Hypothesis Testing: Errors and Power

Whenever we reach a decision with a statistical test, there is always a chance that our decision is the wrong one.

Reality of Hypothesis Testing

H0 is True: There is no difference H0 is False: There is a difference

Observation and Decision Outcomes

H0 is retained Data suggest no difference Correct Decision Type II Error (B = 1 - power) Type II error (false negative Type II error: The error of NOT rejecting the H0 hypothesis when it is false You're not pregnant H0 is rejected Data suggest a difference Type I Error (α) Correct Decision

Power in Hypothesis Testing

Power is the probability of correctly rejecting a false H0 hypothesis, thus good power decreases the likelihood of making Type II error. Memo: It is never possible to prove that a hypothesis is true

QFMP - Jekaterina Rogaten 9 Type I error (false positive)

Statistical Hypothesis Testing: Calculation Steps

• To calculate p(data/H0) for an experiment we need to complete the following steps:
• 1. Identify the sample space = the set of all possible outcomes of the experiment
• 2. Identify an appropriate test statistic that is a function of the sample space
• 3. Define a sampling distribution of the test statistic that assigns a probability to all possible values of the test statistic
• 4. Check if the observed value of the test statistic has a probability smaller than x

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Example: Coin Tossing Experiment

The research hypothesis states that: my friend can make a coin fall as head i.e., when the coin is tossed it is biased towards heads H0 : the probability of success (head) p(s) = .5 H1: the probability of success (head) p(s) > .5 We need to have a situation where some possible outcomes were unlikely to happen by chance The research hypothesis is tested by: Tossing the coin n times Recording the outcome of each trial (s vs. f)

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Sample Space in Coin Tossing

Possible Outcomes Trial 1 S f Trial 2 S f S s f Trial 3 S f s f S s f s f Thus, the sample spaces for 1, 2, and 3 trials are: 1 trial: s, f 2 trials: [s-s, s-f, f-s, f-f] 3 trials: [s-s-s, s-s-f, s-f-s, s-f-f, f-s-s, f-s-f, f-f-s, f-f-f]

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Test Statistic for Coin Tossing

An appropriate test statistic for this experiment is the total number (s) of successes over (n) trials: If the success rate is about 50% of trials, then the coin is likely to be unbiased If the success rate is greater than 50% of trials, then the coin is likely to be biased toward head If the success rate is smaller than 50% of trials, then the coin is likely to be biased toward tail

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Sampling Distribution: 1 Trial

If 1 trial is conducted, what is the probability of obtaining 0 or 1 successes? There are 2 possible outcomes [s, f] Based on the null hypothesis, we regard them as equally probable Therefore, the sampling distribution is p(s = 0) = p(f) = . 5 p(s = 1) = p(s) =. 5

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Sampling Distribution: 2 Trials

. If 2 trials are conducted, what is the probability of obtaining 0, 1, or 2 successes? There are 4 possible outcomes [s-s, s-f, f-s, f-f] Based on the null hypothesis, we regard them as equally probable Therefore, the sampling distribution is: p(s = 0) = p(f,f) = . 25 p(s = 1) = p(s,f) + p(f,s) = . 25 + .25 = . 5 p(s =2) = p(s,s) = . 25

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Sampling Distribution: Binomial Distribution

The sampling distribution of s over n trials is called Binomial distribution The Binomial distribution specifies the probability of each of the possible values of s over n trials: p(s = 0), p(s = 1), p(s =2), ... , p(s = n) For any number of trials, the Binomial distribution can be derived by identifying all of the possible outcomes and summing up their probabilities The next slides show the Binomial distribution for the cases of n = 1 through 8

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Sampling Distribution for n=1

p =. 5 p =. 5 (p =. 5, n=1) Probability 0.5 0.4 0.3 No value of s could lead to rejection of the null hypothesis 0.2 0.1 0 1 0 Number of Successes over 1 Trial

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Sampling Distribution for n=2

p =. 5 (p =. 5, n=2) 0.5 0.4 p =. 25 No value of s could lead to rejection of the null hypothesis 0.2 0.1 0 1 2 0 Number of Successes-over 2-Trials

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Probability p =. 25 0.3

Sampling Distribution for n=3

p =. 375 p =. 375 0.4 (p =. 5, n=3) 0.35 Probability 0.3 0.25 No value of s could lead to rejection of the null hypothesis 0.2 p =. 125 p =. 125 0.15 0.1 0.05 0 1 2 3 0 Number of Successes over 3 Trials

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Sampling Distribution for n=4

0.4 p =. 375 0.35 (p =. 5, n=4) 0.3 p =. 25 p =. 25 0.25 0.2 No value of s could lead to rejection of the null hypothesis 0.15 0.1 p =. 0625 p =. 0625 0.05 0 0 1 2 3 4 Number of Successes over 4 Trials

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Probability

Sampling Distribution for n=5

p(s = 5) <. 05 Rejection region of the null hypothesis 0.35 p =. 313 p =. 313 0.3 Probability 0.25 0.2 p =. 156 p =. 156 (p =. 5, n=5) 0.15 0.1 p =. 031 p =. 031 0.05 0 1 2 3 4 5 0 Number of Successes over 5 Trials

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Sampling Distribution for n=6

p(s =6) <. 05 Rejection region of the null hypothesis 0.35 p =. 313 0.3 p =. 234 p =. 234 0.25 0.2 (p =. 5, n=6) 0.15 p =. 094 p =. D94 0.1 0.05 p =. 016 p =. 016 0 1 2 3 4 5 6 0 Number of Successes over 6 Trials

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Probability

Sampling Distribution for n=7

p(s =7) <. 05 Rejection region of the null hypothesis 0.3 p =. 273 p =. 273 0.25 Probability 0.2 p =. 164 p =. 164 0.15 (p =. 5, n=7) 0.1 p =. 055 p =. 055 0.05 p =. 008 p =. 008 0 1 2 3 4 5 6 7 0 Number of Successes over 7 Trials

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Sampling Distribution for n=8

p(s ≥ 7) <. 05 Rejection region of the null hypothesis 0.3 p =. 273 0.25 p =. 210 p =. 210 0.2 0.15 p =. 109 p= 109 (p =. 5, n=8) 0.1 p =. 031 p =. 031 0.05 p =. 004 p =. 004 0 1 2 3 4 5 6 7 8 0 Probability I Number of Successes over 8 Trials

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