.bg-main4.column[.vmiddle.content[
.amber[Aly Lamuri]  
Indonesia Medical Education and Research Institute
]]

---

---

.bg-main1.column[.vmiddle.content[
- .amber[Unpaired test]
- Paired test
- Final thoughts
- Conceptual remarks
- Case examples
]]

---

# Unpaired test

.font2[
- Kruskal-Wallis H Test
- Differences in multiple groups
- Analogous to one-way ANOVA (.amber[not] its alternative!)
]

???

- ANOVA is not sensitive to non-normality, unless the data is highly skewed
- Kruskal-Wallis is often overused in that regard
- Kruskal-Wallis is somewhat limited as you cannot assign multiple independent
  variables nor adjusting for covariates

---

# Assumptions and limitations

???

- Just like ANOVA, Kruskal-Wallis requires a homogeneous intergroup variances
- When the data presents with a heterogeneous intergroup variance,
  Kruskal-Wallis performs worse than ANOVA
- In such a case, please consider using Welch's ANOVA (`oneway.test` in `R`)
- Welch's method is not available for factorial ANOVA

---

# Procedure

.font2[
- Pool and sort all data element
- Assign rank to sorted data
- Adjust ranks on tied data
- Calculate H statistics
]

.font2[
$$ H = \left[ \frac{12}{\color{orange}{n}(\color{orange}{n}+1)} \displaystyle \sum_{i=1}^\color{red}{k} \frac{\color{yellow}{R}_i^2}{\color{orange}{n}_i} \right] - 3 (\color{orange}{n}+1) $$
]

???

- Only need to understand three arguments
- `$n$`: Total observed value
- `$k$`: Total number of groups
- `$R$`: Rank from pooled data

---

# Example, please?

---

```r
# CO2 dataset in R
str(DNase)
```

```
## Classes 'nfnGroupedData', 'nfGroupedData', 'groupedData' and 'data.frame':	176 obs. of  3 variables:
##  $ Run    : Ord.factor w/ 11 levels "10"<"11"<"9"<..: 4 4 4 4 4 4 4 4 4 4 ...
##  $ conc   : num  0.0488 0.0488 0.1953 0.1953 0.3906 ...
##  $ density: num  0.017 0.018 0.121 0.124 0.206 0.215 0.377 0.374 0.614 0.609 ...
##  - attr(*, "formula")=Class 'formula'  language density ~ conc | Run
##   .. ..- attr(*, ".Environment")=<environment: R_EmptyEnv> 
##  - attr(*, "labels")=List of 2
##   ..$ x: chr "DNase concentration"
##   ..$ y: chr "Optical density"
##  - attr(*, "units")=List of 1
##   ..$ x: chr "(ng/ml)"
```

???

- We will use `DNase` dataset
- It is a result obtained from an ELISA assay 
- `Run`: The assay run
- `conc`: Protein concentration
- `density`: Optical density in the assay

---

```r
with(DNase, tapply(density, Run, shapiro.test)) %>%
	lapply(broom::tidy) %>% lapply(data.frame) %>% {do.call(rbind, .)} %>%
	knitr::kable() %>% kable_minimal()
```

<table class=" lightable-minimal" style='font-family: "Trebuchet MS", verdana, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:left;"> method </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 10 </td>
   <td style="text-align:right;"> 0.891 </td>
   <td style="text-align:right;"> 0.059 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 11 </td>
   <td style="text-align:right;"> 0.888 </td>
   <td style="text-align:right;"> 0.051 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 9 </td>
   <td style="text-align:right;"> 0.889 </td>
   <td style="text-align:right;"> 0.053 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:right;"> 0.883 </td>
   <td style="text-align:right;"> 0.044 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 4 </td>
   <td style="text-align:right;"> 0.877 </td>
   <td style="text-align:right;"> 0.035 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 8 </td>
   <td style="text-align:right;"> 0.876 </td>
   <td style="text-align:right;"> 0.033 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 5 </td>
   <td style="text-align:right;"> 0.879 </td>
   <td style="text-align:right;"> 0.037 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 7 </td>
   <td style="text-align:right;"> 0.883 </td>
   <td style="text-align:right;"> 0.043 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 6 </td>
   <td style="text-align:right;"> 0.880 </td>
   <td style="text-align:right;"> 0.039 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:right;"> 0.869 </td>
   <td style="text-align:right;"> 0.027 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:right;"> 0.880 </td>
   <td style="text-align:right;"> 0.039 </td>
   <td style="text-align:left;"> Shapiro-Wilk normality test </td>
  </tr>
</tbody>
</table>

---

```r
with(DNase, car::leveneTest(conc, Run))
```

```
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group  10       0      1
##       165
```

---

```r
kruskal.test(conc ~ Run, data=DNase)
```

```
## 
## 	Kruskal-Wallis rank sum test
## 
## data:  conc by Run
## Kruskal-Wallis chi-squared = 0, df = 10, p-value = 1
```

```r
rstatix::kruskal_effsize(conc ~ Run, data=DNase)
```

```
## # A tibble: 1 x 5
##   .y.       n effsize method  magnitude
## * <chr> <int>   <dbl> <chr>   <ord>    
## 1 conc    176 -0.0606 eta2[H] moderate
```

---

```r
dunn.test::dunn.test(DNase$conc, DNase$Run)
```

```
##   Kruskal-Wallis rank sum test
## 
## data: x and group
## Kruskal-Wallis chi-squared = 0, df = 10, p-value = 1
## 
## 
##                            Comparison of x by group                            
##                                 (No adjustment)                                
## Col Mean-|
## Row Mean |          1         10         11          2          3          4
## ---------+------------------------------------------------------------------
##       10 |   0.000000
##          |     0.5000
##          |
##       11 |   0.000000   0.000000
##          |     0.5000     0.5000
##          |
##        2 |   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000
##          |
##        3 |   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000
##          |
##        4 |   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        5 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        6 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        7 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        8 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        9 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
## Col Mean-|
## Row Mean |          5          6          7          8
## ---------+--------------------------------------------
##        6 |   0.000000
##          |     0.5000
##          |
##        7 |   0.000000   0.000000
##          |     0.5000     0.5000
##          |
##        8 |   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000
##          |
##        9 |   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000
## 
## alpha = 0.05
## Reject Ho if p <= alpha/2
```

---

.bg-main1.column[.vmiddle.content[
- Unpaired test
- .amber[Paired test]
- Final thoughts
- Conceptual remarks
- Case examples
]]

---

# Paired test

.font2[
- Analogous to repeated measure ANOVA
- Calculate .amber[*within*] subject differences
- Compare .amber[*between*] group differences  
]

---

# Procedure

.font2[
- Rank observation in the same subjects
- Sum all ranks within the same group
- Calculate the statistics
]

.font2[
$$ Q = \left[ \frac{12 \color{red}{N}}{\color{red}{N} \color{yellow}{k} ( \color{yellow}{k}+1)} \displaystyle \sum_{i=1}^{\color{yellow}{k}} \color{orange}{R}_i^2 \right] - 3 \color{red}{N}( \color{yellow}{k}+1) $$
]

???

- `$N$`: Number of rows (block)
- `$k$`: Number of columns (treatment / repetition)
- `$R$`: Ranked values

---

# Example, please?

---

```r
str(warpbreaks)
```

```
## 'data.frame':	54 obs. of  3 variables:
##  $ breaks : num  26 30 54 25 70 52 51 26 67 18 ...
##  $ wool   : Factor w/ 2 levels "A","B": 1 1 1 1 1 1 1 1 1 1 ...
##  $ tension: Factor w/ 3 levels "L","M","H": 1 1 1 1 1 1 1 1 1 2 ...
```

---

```r
wp <- aggregate(warpbreaks$breaks,
	by = list(
		w = warpbreaks$wool,
		t = warpbreaks$tension
	), FUN = mean
)
friedman.test(x ~ w | t, data=wp)
```

```
## 
## 	Friedman rank sum test
## 
## data:  x and w and t
## Friedman chi-squared = 0.3, df = 1, p-value = 0.6
```

```r
rstatix::friedman_effsize(x ~ w | t, data=wp)
```

```
## # A tibble: 1 x 5
##   .y.       n effsize method    magnitude
## * <chr> <int>   <dbl> <chr>     <ord>    
## 1 x         3   0.111 Kendall W small
```

---

.bg-main1.column[.vmiddle.content[
- Unpaired test
- Paired test
- .amber[Final thoughts]
- Conceptual remarks
- Case examples
]]

???

- We have learnt most of the basics for statistical test, both in parametric
  and non-parametric approaches
- We may need to contemplate what we have learnt so far

---

# Excerpts on non-parametric test

.font2[
- Limited if compared to parametric tests
- Whenever possible, use parametric tests
- However, the non-parametric test is better in ordinal data
]

???

- In the case of having an ordinal data as your dependent variable, parametric
  test is practically unusable
- We use parametric tests to handle numeric data

---

# Parametric test with non-normal data?

.font2[
- You may consider this approach
- Need a substantially large sample size
- Please be careful with skewed data
- .amber[Homogeneity] of intergroup variances is an important assumption!
]

---

# Further analysis after ANOVA

.font2[
- Post-hoc analysis is a must
- ANOVA is an explanatory statistical model
- Residual analysis to test model goodness of fit
- Assumption:
  - .amber[Residual] normality
  - Homogeneity of .amber[residual] variances
]

???

- In residual analyses, we need to satisfy both assumptions
- You can check normality using a QQ-plot or statistical test
- Homogeneous residual variance is homoscedasticity
- Test for homoscedasticity: Breusch-Pagan or Harrison-McCabe test

---

.bg-main1.column[.vmiddle.content[
- Unpaired test
- Paired test
- Final thoughts
- .amber[Conceptual remarks]
- Case examples
]]

---

# Conceptual remarks

.font2[
- Distribution: discrete and continuous
- Hypotheses: `$H_0$` and `$H_1$`
- Statistical tests
- Independent and dependent variables
]

???

- Mention about binomial, Poisson, normal, and `$\chi^2$` distribution
- Explain more on differences between independent and dependent variables

---

# What test should we use?

.font2[
- Categoric DV and categoric IV
- Numeric DV and categoric IV (2 groups)
- Numeric DV and categoric IV (>2 groups)
]

???

- How about numeric DV and numeric IV?
- And how if we have categoric DV and numeric IV?

---

.bg-main1.column[.vmiddle.content[
- Unpaired test
- Paired test
- Final thoughts
- Conceptual remarks
- .amber[Case examples]
]]

Notes for current slide

Notes for next slide

Differences in Multiple Groups

Aly Lamuri
Indonesia Medical Education and Research Institute

1 / 11

Overview

Unpaired test
Paired test
Final thoughts
Conceptual remarks
Case examples

1 / 11

Unpaired testKruskal-Wallis H Test
Differences in multiple groups
Analogous to one-way ANOVA (not its alternative!)

1 / 11

ANOVA is not sensitive to non-normality, unless the data is highly skewed
Kruskal-Wallis is often overused in that regard
Kruskal-Wallis is somewhat limited as you cannot assign multiple independent variables nor adjusting for covariates

Assumptions and limitationsI.I.D
Does not assume normality
Requires homogeneous intergroup variances

1 / 11

Just like ANOVA, Kruskal-Wallis requires a homogeneous intergroup variances
When the data presents with a heterogeneous intergroup variance, Kruskal-Wallis performs worse than ANOVA
In such a case, please consider using Welch's ANOVA (oneway.test in R)
Welch's method is not available for factorial ANOVA

ProcedurePool and sort all data element
Assign rank to sorted data
Adjust ranks on tied data
Calculate H statistics

2 / 11

Procedure

Pool and sort all data element
Assign rank to sorted data
Adjust ranks on tied data
Calculate H statistics

$H = [\frac{12}{n (n + 1)} \sum_{i = 1}^{k} \frac{R_{i}^{2}}{n_{i}}] - 3 (n + 1)$

2 / 11

Only need to understand three arguments
$n$ : Total observed value
$k$ : Total number of groups
$R$ : Rank from pooled data

Procedure

Pool and sort all data element
Assign rank to sorted data
Adjust ranks on tied data
Calculate H statistics

$H = [\frac{12}{n (n + 1)} \sum_{i = 1}^{k} \frac{R_{i}^{2}}{n_{i}}] - 3 (n + 1)$

$H \sim χ^{2} (k - 1)$

2 / 11

Only need to understand three arguments
$n$ : Total observed value
$k$ : Total number of groups
$R$ : Rank from pooled data

Example, please?

# CO2 dataset in R
str(DNase)

## Classes 'nfnGroupedData', 'nfGroupedData', 'groupedData' and 'data.frame':    176 obs. of  3 variables:
##  $ Run    : Ord.factor w/ 11 levels "10"<"11"<"9"<..: 4 4 4 4 4 4 4 4 4 4 ...
##  $ conc   : num  0.0488 0.0488 0.1953 0.1953 0.3906 ...
##  $ density: num  0.017 0.018 0.121 0.124 0.206 0.215 0.377 0.374 0.614 0.609 ...
##  - attr(*, "formula")=Class 'formula'  language density ~ conc | Run
##   .. ..- attr(*, ".Environment")=<environment: R_EmptyEnv> 
##  - attr(*, "labels")=List of 2
##   ..$ x: chr "DNase concentration"
##   ..$ y: chr "Optical density"
##  - attr(*, "units")=List of 1
##   ..$ x: chr "(ng/ml)"

3 / 11

We will use DNase dataset
It is a result obtained from an ELISA assay
Run: The assay run
conc: Protein concentration
density: Optical density in the assay

Example, please?

with(DNase, tapply(density, Run, shapiro.test)) %>%
    lapply(broom::tidy) %>% lapply(data.frame) %>% {do.call(rbind, .)} %>%
    knitr::kable() %>% kable_minimal()

	statistic	p.value	method
10	0.891	0.059	Shapiro-Wilk normality test
11	0.888	0.051	Shapiro-Wilk normality test
9	0.889	0.053	Shapiro-Wilk normality test
1	0.883	0.044	Shapiro-Wilk normality test
4	0.877	0.035	Shapiro-Wilk normality test
8	0.876	0.033	Shapiro-Wilk normality test
5	0.879	0.037	Shapiro-Wilk normality test
7	0.883	0.043	Shapiro-Wilk normality test
6	0.880	0.039	Shapiro-Wilk normality test
2	0.869	0.027	Shapiro-Wilk normality test
3	0.880	0.039	Shapiro-Wilk normality test

3 / 11

Example, please?

with(DNase, car::leveneTest(conc, Run))

## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group  10       0      1
##       165

3 / 11

Example, please?

kruskal.test(conc ~ Run, data=DNase)

## 
##     Kruskal-Wallis rank sum test
## 
## data:  conc by Run
## Kruskal-Wallis chi-squared = 0, df = 10, p-value = 1

rstatix::kruskal_effsize(conc ~ Run, data=DNase)

## # A tibble: 1 x 5
##   .y.       n effsize method  magnitude
## * <chr> <int>   <dbl> <chr>   <ord>    
## 1 conc    176 -0.0606 eta2[H] moderate

3 / 11

Example, please?

dunn.test::dunn.test(DNase$conc, DNase$Run)

##   Kruskal-Wallis rank sum test
## 
## data: x and group
## Kruskal-Wallis chi-squared = 0, df = 10, p-value = 1
## 
## 
##                            Comparison of x by group                            
##                                 (No adjustment)                                
## Col Mean-|
## Row Mean |          1         10         11          2          3          4
## ---------+------------------------------------------------------------------
##       10 |   0.000000
##          |     0.5000
##          |
##       11 |   0.000000   0.000000
##          |     0.5000     0.5000
##          |
##        2 |   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000
##          |
##        3 |   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000
##          |
##        4 |   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        5 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        6 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        7 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        8 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
##          |
##        9 |   0.000000   0.000000   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000     0.5000     0.5000
## Col Mean-|
## Row Mean |          5          6          7          8
## ---------+--------------------------------------------
##        6 |   0.000000
##          |     0.5000
##          |
##        7 |   0.000000   0.000000
##          |     0.5000     0.5000
##          |
##        8 |   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000
##          |
##        9 |   0.000000   0.000000   0.000000   0.000000
##          |     0.5000     0.5000     0.5000     0.5000
## 
## alpha = 0.05
## Reject Ho if p <= alpha/2

3 / 11

Overview

Unpaired test
Paired test
Final thoughts
Conceptual remarks
Case examples

3 / 11

Paired testAnalogous to repeated measure ANOVA
Calculate within subject differences
Compare between group differences  

4 / 11

ProcedureRank observation in the same subjects
Sum all ranks within the same group
Calculate the statistics

5 / 11

Procedure

Rank observation in the same subjects
Sum all ranks within the same group
Calculate the statistics

$Q = [\frac{12 N}{N k (k + 1)} \sum_{i = 1}^{k} R_{i}^{2}] - 3 N (k + 1)$

5 / 11

$N$ : Number of rows (block)
$k$ : Number of columns (treatment / repetition)
$R$ : Ranked values

Procedure

Rank observation in the same subjects
Sum all ranks within the same group
Calculate the statistics

$Q = [\frac{12 N}{N k (k + 1)} \sum_{i = 1}^{k} R_{i}^{2}] - 3 N (k + 1)$

$Q \sim χ^{2} (k - 1)$

5 / 11

$N$ : Number of rows (block)
$k$ : Number of columns (treatment / repetition)
$R$ : Ranked values

Example, please?

str(warpbreaks)

## 'data.frame':    54 obs. of  3 variables:
##  $ breaks : num  26 30 54 25 70 52 51 26 67 18 ...
##  $ wool   : Factor w/ 2 levels "A","B": 1 1 1 1 1 1 1 1 1 1 ...
##  $ tension: Factor w/ 3 levels "L","M","H": 1 1 1 1 1 1 1 1 1 2 ...

6 / 11

Example, please?

wp <- aggregate(warpbreaks$breaks,
    by = list(
        w = warpbreaks$wool,
        t = warpbreaks$tension
    ), FUN = mean
)
friedman.test(x ~ w | t, data=wp)

## 
##     Friedman rank sum test
## 
## data:  x and w and t
## Friedman chi-squared = 0.3, df = 1, p-value = 0.6

rstatix::friedman_effsize(x ~ w | t, data=wp)

## # A tibble: 1 x 5
##   .y.       n effsize method    magnitude
## * <chr> <int>   <dbl> <chr>     <ord>    
## 1 x         3   0.111 Kendall W small

6 / 11

Overview

Unpaired test
Paired test
Final thoughts
Conceptual remarks
Case examples

6 / 11

We have learnt most of the basics for statistical test, both in parametric and non-parametric approaches
We may need to contemplate what we have learnt so far

Excerpts on non-parametric testLimited if compared to parametric tests
Whenever possible, use parametric tests
However, the non-parametric test is better in ordinal data

7 / 11

In the case of having an ordinal data as your dependent variable, parametric test is practically unusable
We use parametric tests to handle numeric data

Parametric test with non-normal data?You may consider this approach
Need a substantially large sample size
Please be careful with skewed data
Homogeneity of intergroup variances is an important assumption!

8 / 11

Further analysis after ANOVAPost-hoc analysis is a must
ANOVA is an explanatory statistical model
Residual analysis to test model goodness of fit
Assumption:Residual normality
Homogeneity of residual variances


9 / 11

In residual analyses, we need to satisfy both assumptions
You can check normality using a QQ-plot or statistical test
Homogeneous residual variance is homoscedasticity
Test for homoscedasticity: Breusch-Pagan or Harrison-McCabe test

Overview

Unpaired test
Paired test
Final thoughts
Conceptual remarks
Case examples

9 / 11

Conceptual remarksDistribution: discrete and continuous
Hypotheses: H0H0 and H1H1
Statistical tests
Independent and dependent variables

10 / 11

Mention about binomial, Poisson, normal, and $χ^{2}$ distribution
Explain more on differences between independent and dependent variables

What test should we use?Categoric DV and categoric IV
Numeric DV and categoric IV (2 groups)
Numeric DV and categoric IV (>2 groups)

11 / 11

How about numeric DV and numeric IV?
And how if we have categoric DV and numeric IV?

Overview

Unpaired test
Paired test
Final thoughts
Conceptual remarks
Case examples

11 / 11

Overview

Unpaired test
Paired test
Final thoughts
Conceptual remarks
Case examples

1 / 11

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