Martijn Wieling

University of Groningen

- Three variants of the \(t\)-test
- How to calculate the effect size (Cohen's \(d\))
- How to report results of a statistical test

- Nonparametric tests:
- Mann-Whitney U test: alternative to independent samples \(t\)-test
- Wilcoxon signed-rank test: alternative to paired and single sample \(t\)-test
- Sign test: alternative to Wilcoxon signed-rank test

- Reporting statistical analyses (again)

- Nonparametric tests do
**not**assume an underlying distribution (of the data) and therefore have no parameters- In contrast to e.g., \(N(0,1)\) and \(t(18)\)

- Nonparametric tests are applied when the distribution is unknown or the required assumptions of the parametric test are violated
- They can also be applied to data assumed to be normally distributed

- Often best option for nonnumeric data (next lecture: \(\chi^2\))
- Less sensitive than parametric tests (i.e. less
**power**)!

**Mann-Whitney U test**: alternative to independent samples \(t\)-test- When data normally distributed: 95% of power of \(t\)-test

**Wilcoxon signed-rank test**: alternative to paired \(t\)-test- Requirement: distribution symmetrical
- When data normally distributed: 95% of power of \(t\)-test

**Sign test**: alternative to Wilcoxon signed-rank test when data not symmetrical

- Alternative to independent samples \(t\)-test (i.e. comparing two indep. samples)
- Applicable to ordinal data (there is an ordering: no exact scale) and num. data
- Also when \(n \leq\) 30 and data in (at least) one group not normally distributed
- \(H_0\): \(P(X > Y) = P(Y > X)\), \(H_a\): \(P(X > Y) \neq P(Y > X)\)
- If distributions of samples the same, this also means:

\(H_0\): medians of both groups equal, \(H_a\): medians of both groups differ

- If distributions of samples the same, this also means:

- Frequently applied to Likert data: on a scale from 1 (easiest) to 5 (hardest) ...
- (Identical to: Wilcoxon's
**rank sum**test)

- Idea: combine the two sets of values, order them from low to high and count how often the items in one set come after items in the other set
- Ties add 0.5 (instead of 1) to the counts

- Group A: (2, 4, 6, 10, 20), Group B: (8, 12, 14, 16, 18)
- Ordered: A A A B A B B B B A (values: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20)
- \(U_A = 0 + 0 + 0 + 1 + 5 = 6\) (A)
- \(U_B = 3 + 4 + 4 + 4 + 4 = 19\) (B)

- Mann-Whitney U \(= min(U_A,U_B) = 6\)
- The lower U, the more likely to be significantly different

- \(p\)-value: obtained by assessing where \(U\) is located in the distribution of all possible \(U\)-values for given sample sizes \(n_A\) and \(n_B\)
- Distribution of all \(U\)-values resembles a
**normal distribution**(for larger \(n\))

- Distribution of all \(U\)-values resembles a

- In
`R`

:`wilcox.test()`

- Identical usage as
`t.test()`

- Identical usage as
- With Mann-Whitney U test: data is converted to ranks
- Actual values are ignored: loss of information!

- Research question: Do native English speakers show a stronger distinction of /t/ from /θ/ ("th") with their tongue than non-native (Dutch) speakers of English?
- Hypothesis: The tongue position difference between /t/ and /θ/ is larger for English native speakers than for non-native Dutch speakers of English
- \(H_0\): same frontal /t/-/θ /position difference for Dutch and English speakers
- \(H_a\): larger frontal /t/-/θ position difference for English (versus Dutch) speakers

- Data: 22 English and 19 Dutch participants who pronounced 10 minimal pairs

/t/:/θ/ while connected to the articulography device:- 'fate'-'faith', 'forth'-'fort', 'kit'-'kith', 'mitt'-'myth', 'tent'-'tenth'
- 'tank'-'thank', 'team'-'theme', 'tick'-'thick', 'ties'-'thighs', 'tongs'-'thongs'

- For each speaker, we calculated the average difference in frontal tongue position between /t/-words and /θ/-words

- As the values in one group are not normally distributed, we used the
**Mann-Whitney U test**to analyze the difference between the two groups - Our \(\alpha\)-level is set at 0.05 (one-tailed)

```
wilcox.test(diffEN$Diff, diffNL$Diff, alternative = "greater") # 1st > 2nd?
```

```
#
# Wilcoxon rank sum exact test
#
# data: diffEN$Diff and diffNL$Diff
# W = 315, p-value = 0.0025
# alternative hypothesis: true location shift is greater than 0
```

- We reject the null hypothesis, and accept the alternative hypothesis that the native English speakers show a greater tongue distinction between /t/ and /θ/ than non-native speakers (U: 315, \(p = 0.0025\))
- If we would have incorrectly analyzed the data using the independent samples \(t\)-test, we would also have rejected the null hypothesis
- But with \(p = 0.004\)

- Cliff's delta (or \(d\)) measures effect size of the Mann-Whitney U test
- \(|d| < 0.147\): negl.; \(|d| < 0.33\): small; \(|d| < 0.474\): medium; \(|d| \geq 0.474\): large

```
library(effsize)
cliff.delta(diffEN$Diff, diffNL$Diff)
```

```
#
# Cliff's Delta
#
# delta estimate: 0.50718 (large)
# 95 percent confidence interval:
# lower upper
# 0.13076 0.75579
```

- Instead of the Mann-Whitney U test, an independent samples \(t\)-test of the ranks gives a \(p\)-value close to that of the Mann-Whitney U test
- E.g., A A A B A B B B B A: ranks group A = (1,2,3,5,10), ranks group B: (4,6,7,8,9)
- For our example: \(p = 0.0024\) (Mann-Whitney U test: \(p = 0.0025\))

- Mann-Whitney U test cannot be applied to single samples, nor paired data
- For that we use the
**Wilcoxon signed-rank test**

- For that we use the

- Alternative to single sample or paired \(t\)-test
- Applied when data is non-normal
- However, distribution should be roughly symmetric, not skewed
- If distribution is skewed,
**sign test**should be used

- Applicable to ordinal and numerical data

- Applied when data is non-normal

- For paired samples:
- \(H_0\): median of the differences \(=\) 0
- \(H_a\): median of the differences \(\neq\) 0 (for two-tailed hypothesis)

- For single sample:
- \(H_0\): distribution symmetric around \(x\) (\(\approx \mu = x\), due to symmetry)
- \(H_a\): distribution non-symmetric around \(x\) (\(\approx \mu \neq x\))
- If \(H_0\) rejected: results may be reported as being significantly different from \(x\)

- Calculate pairwise differences (single sample: with respect to single value)
- Rank the
**absolute**differences from low to high (excluding differences of 0) - Add the signs of the differences to the ranks
- Sum the positive ranks: \(W\)
- If \(H_0\) true then \(W\) close to half of the total sum of all unsigned-ranks

- \(p\)-value: obtained by assessing where \(W\) is located in the distribution of all possible \(W\)-values for a given sample size \(n\)
- Distribution of all \(W\)-values resembles a
**normal distribution**(for larger \(n\))

- Distribution of all \(W\)-values resembles a

- Example: comparing English scores to 7.5 (only 6 cases)

english_score | diff | abs_diff | rank | signed_rank |
---|---|---|---|---|

6.78 | -0.72 | 0.72 | 3 | -3 |

5.94 | -1.56 | 1.56 | 6 | -6 |

6.18 | -1.32 | 1.32 | 5 | -5 |

8.00 | 0.50 | 0.50 | 2 | 2 |

7.42 | -0.08 | 0.08 | 1 | -1 |

8.63 | 1.13 | 1.13 | 4 | 4 |

- \(W = 2 + 4 = 6\)
- Compared to half of total sum of ranks (\(21 / 2 = 10.5\), so quite different)

- Fortunately we don't have to do this manually!
- In
`R`

:`wilcox.test()`

(same as for Mann-Whitney U) - Data is converted to ranks: actual values are ignored (i.e. information loss)

- Given our English proficiency data, we'd like to assess if the average English score is different from 7.5 (with \(\alpha = 0.05\))
- \(H_0\): \(\mu = 7.5\)
- \(H_a\): \(\mu \neq 7.5\)

- Visualization: