Directions

Let’s go!!!

1. Independent samples t-test

1.1 Load data

smash <- read.csv("smash_3.csv", header = TRUE)
head(smash)

##     character        game regular_fall fast_fall weight walk_speed run_speed
## 1       mario Super Mario         1.50     2.400     98      1.155     1.760
## 2 donkey kong Donkey Kong         1.63     2.608    127      1.365     1.873
## 3        link       Zelda         1.60     3.040    104      1.247     1.534
## 4       samus     Metroid         1.33     2.168    108      1.115     1.654
## 5  dark samus     Metroid         1.33     2.168    108      1.115     1.654
## 6       yoshi Super Mario         1.29     2.064    104      1.208     2.046
##   full_hop_height short_hop_height double_hop_height before after expertise
## 1           36.33            17.54             36.33      3     8   Amateur
## 2           34.00            17.30             35.50      8    10    Expert
## 3           27.80            13.38             29.00     10     7    Expert
## 4           37.00            18.00             37.00      6     9    Expert
## 5           37.00            18.00             37.00      2     5   Amateur
## 6           36.09            14.43             51.56      5     9   Amateur

table(smash$game)

## 
## Animal Crossing   Banjo Kazooie       Bayonetta     Castlevania     Donkey Kong 
##               2               1               1               2               3 
##    Dragon Quest       Duck Hunt      Earthbound          F-Zero   Final Fantasy 
##               1               1               1               2               1 
##     Fire Emblem          Icarus           Kirby        Mega Man      Metal Gear 
##               6               2               4               1               1 
##         Metroid          Mother        Nintendo          Pikmin         Pokemon 
##               4               1               1               1               7 
##       Punch Out           Sonic        Splatoon        Stack Up        Star Fox 
##               1               1               1               1               2 
##  Street Fighter     Super Mario             Wii       Xenoblade           Zelda 
##               2              10               4               1               6

1.2 Independent samples t-test

H0: Metroid = Zelda

H1: Metroid <> Zelda

Subset required data. Compare Metroid vs. Zelda.

subset_1 <- subset(smash, game == "Metroid" | game == "Zelda")
head(subset_1)

##     character    game regular_fall fast_fall weight walk_speed run_speed
## 3        link   Zelda         1.60     3.040    104      1.247     1.534
## 4       samus Metroid         1.33     2.168    108      1.115     1.654
## 5  dark samus Metroid         1.33     2.168    108      1.115     1.654
## 17      sheik   Zelda         1.75     2.800     78      1.470     2.420
## 18      zelda   Zelda         1.35     2.160     85      0.914     1.430
## 23 young link   Zelda         1.80     2.880     88      1.260     1.749
##    full_hop_height short_hop_height double_hop_height before after expertise
## 3            27.80            13.38             29.00     10     7    Expert
## 4            37.00            18.00             37.00      6     9    Expert
## 5            37.00            18.00             37.00      2     5   Amateur
## 17           39.00            18.75             40.00      6     5    Expert
## 18           31.55            15.24             31.55      4     7   Amateur
## 23           33.66            16.26             33.66      7    10    Expert

Assuming unequal variance by default.

t.test(subset_1$run_speed ~ subset_1$game)

## 
##  Welch Two Sample t-test
## 
## data:  subset_1$run_speed by subset_1$game
## t = 0.94211, df = 7.1841, p-value = 0.3767
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.3363160  0.7856494
## sample estimates:
## mean in group Metroid   mean in group Zelda 
##              1.954500              1.729833

Assuming equal variance.

t.test(subset_1$run_speed ~ subset_1$game, var.equal = TRUE)

## 
##  Two Sample t-test
## 
## data:  subset_1$run_speed by subset_1$game
## t = 0.91593, df = 8, p-value = 0.3865
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.3409716  0.7903049
## sample estimates:
## mean in group Metroid   mean in group Zelda 
##              1.954500              1.729833

1.3 Assumptions

Test for normality.

shapiro.test(subset_1$run_speed)

## 
##  Shapiro-Wilk normality test
## 
## data:  subset_1$run_speed
## W = 0.92509, p-value = 0.4013

2. One direction t-test

2.1 Subset data

Compare Kirby vs. Super Mario.

subset_2 <- subset(smash, game == "Kirby" | game == "Super Mario")
head(subset_2)

##    character        game regular_fall fast_fall weight walk_speed run_speed
## 1      mario Super Mario         1.50     2.400     98      1.155     1.760
## 6      yoshi Super Mario         1.29     2.064    104      1.208     2.046
## 7      kirby       Kirby         1.23     1.968     79      0.977     1.727
## 10     luigi Super Mario         1.32     2.112     97      1.134     1.650
## 14     peach Super Mario         1.19     1.904     89      0.924     1.595
## 15     daisy Super Mario         1.19     1.904     89      0.924     1.595
##    full_hop_height short_hop_height double_hop_height before after expertise
## 1            36.33            17.54             36.33      3     8   Amateur
## 6            36.09            14.43             51.56      5     9   Amateur
## 7            25.37            12.24             22.00      5     9   Amateur
## 10           44.00            19.98             41.31      7    10    Expert
## 14           30.03            14.50             30.03      2     5   Amateur
## 15           30.03            14.50             30.03     10     8    Expert

2.2 Execute t-test

2.2.1 Less than alternate hypothesis

H0: Kirby >= Super Mario

H1: Kirby < Super Mario

Assuming unequal variance by default.

t.test(subset_2$regular_fall ~ subset_2$game, alternative = "less")

## 
##  Welch Two Sample t-test
## 
## data:  subset_2$regular_fall by subset_2$game
## t = 0.48396, df = 4.7882, p-value = 0.6751
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##       -Inf 0.4319869
## sample estimates:
##       mean in group Kirby mean in group Super Mario 
##                     1.580                     1.497

Assuming equal variance.

t.test(subset_2$regular_fall ~ subset_2$game, alternative = "less", var.equal = TRUE)

## 
##  Two Sample t-test
## 
## data:  subset_2$regular_fall by subset_2$game
## t = 0.52578, df = 12, p-value = 0.6957
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##       -Inf 0.3643548
## sample estimates:
##       mean in group Kirby mean in group Super Mario 
##                     1.580                     1.497

2.2.2 Greater than alternate hypothesis

H0: Kirby <= Super Mario

H1: Kirby > Super Mario

Assuming unequal variance by default.

t.test(subset_2$regular_fall ~ subset_2$game, alternative = "greater")

## 
##  Welch Two Sample t-test
## 
## data:  subset_2$regular_fall by subset_2$game
## t = 0.48396, df = 4.7882, p-value = 0.3249
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  -0.2659869        Inf
## sample estimates:
##       mean in group Kirby mean in group Super Mario 
##                     1.580                     1.497

Assuming equal variance.

t.test(subset_2$regular_fall ~ subset_2$game, alternative = "greater", var.equal = TRUE)

## 
##  Two Sample t-test
## 
## data:  subset_2$regular_fall by subset_2$game
## t = 0.52578, df = 12, p-value = 0.3043
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  -0.1983548        Inf
## sample estimates:
##       mean in group Kirby mean in group Super Mario 
##                     1.580                     1.497

2.3 Assumptions

Test for normality.

shapiro.test(subset_2$regular_fall)

## 
##  Shapiro-Wilk normality test
## 
## data:  subset_2$regular_fall
## W = 0.92898, p-value = 0.2953

3. Paired-samples t-test

Player skill level by character before and after training.

3.1 Paired samples t-test

H0: before = after

H1: before <> after

t.test(smash$before, smash$after, paired = TRUE)

## 
##  Paired t-test
## 
## data:  smash$before and smash$after
## t = -4.4097, df = 71, p-value = 3.606e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -2.0370808 -0.7684747
## sample estimates:
## mean of the differences 
##               -1.402778

3.1 Less than alternate hypothesis

t.test(smash$before, smash$after, paired = TRUE, alternative = "less")

## 
##  Paired t-test
## 
## data:  smash$before and smash$after
## t = -4.4097, df = 71, p-value = 1.803e-05
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##        -Inf -0.8726076
## sample estimates:
## mean of the differences 
##               -1.402778

3.2 Greater than alternate hypothesis

t.test(smash$before, smash$after, paired = TRUE, alternative = "greater")

## 
##  Paired t-test
## 
## data:  smash$before and smash$after
## t = -4.4097, df = 71, p-value = 1
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  -1.932948       Inf
## sample estimates:
## mean of the differences 
##               -1.402778

3.4 Assumptions

Test for normality.

shapiro.test(smash$before)

## 
##  Shapiro-Wilk normality test
## 
## data:  smash$before
## W = 0.92748, p-value = 0.0004692

shapiro.test(smash$after)

## 
##  Shapiro-Wilk normality test
## 
## data:  smash$after
## W = 0.90091, p-value = 3.329e-05

4. F-Test

2-sided test.

H0: Variance of walk_speed = Variance of run_speed

H1: Variance of walk_speed <> Variance of run_speed

var.test(smash$walk_speed, smash$run_speed, 
         alternative = "two.sided")

## 
##  F test to compare two variances
## 
## data:  smash$walk_speed and smash$run_speed
## F = 0.2979, num df = 71, denom df = 71, p-value = 7.777e-07
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  0.1863816 0.4761475
## sample estimates:
## ratio of variances 
##          0.2979012

1-sided upper

H0: Variance of walk_speed <= Variance of run_speed

H1: Variance of walk_speed > Variance of run_speed

var.test(smash$walk_speed, smash$run_speed, 
         alternative = "greater")

## 
##  F test to compare two variances
## 
## data:  smash$walk_speed and smash$run_speed
## F = 0.2979, num df = 71, denom df = 71, p-value = 1
## alternative hypothesis: true ratio of variances is greater than 1
## 95 percent confidence interval:
##  0.2010832       Inf
## sample estimates:
## ratio of variances 
##          0.2979012

1-sided lower

H0: Variance of walk_speed >= Variance of run_speed

H1: Variance of walk_speed < Variance of run_speed

var.test(smash$walk_speed, smash$run_speed, 
         alternative = "less")

## 
##  F test to compare two variances
## 
## data:  smash$walk_speed and smash$run_speed
## F = 0.2979, num df = 71, denom df = 71, p-value = 3.888e-07
## alternative hypothesis: true ratio of variances is less than 1
## 95 percent confidence interval:
##  0.0000000 0.4413354
## sample estimates:
## ratio of variances 
##          0.2979012

4.1 Alternate data format

Subset data.

subset_3 <- smash[, c(1, 6, 7)]
head(subset_3)

##     character walk_speed run_speed
## 1       mario      1.155     1.760
## 2 donkey kong      1.365     1.873
## 3        link      1.247     1.534
## 4       samus      1.115     1.654
## 5  dark samus      1.115     1.654
## 6       yoshi      1.208     2.046

Convert to long.

library(reshape2)
subset_3_long <- melt(subset_3, 
                  id.vars = c("character"),
                  variable.name = "movement",
                  value.name = "speed")
head(subset_3_long)

##     character   movement speed
## 1       mario walk_speed 1.155
## 2 donkey kong walk_speed 1.365
## 3        link walk_speed 1.247
## 4       samus walk_speed 1.115
## 5  dark samus walk_speed 1.115
## 6       yoshi walk_speed 1.208

tail(subset_3_long)

##             character  movement speed
## 139       king k rool run_speed 1.485
## 140          isabelle run_speed 1.480
## 141        incineroar run_speed 1.180
## 142     piranha plant run_speed 1.720
## 143              hero run_speed 1.840
## 144 banjo and kazooie run_speed 2.180

2-sided test.

H0: Variance of walk_speed = Variance of run_speed

H1: Variance of walk_speed <> Variance of run_speed

var.test(subset_3_long$speed ~ subset_3_long$movement, 
         alternative = "two.sided")

## 
##  F test to compare two variances
## 
## data:  subset_3_long$speed by subset_3_long$movement
## F = 0.2979, num df = 71, denom df = 71, p-value = 7.777e-07
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  0.1863816 0.4761475
## sample estimates:
## ratio of variances 
##          0.2979012

1-sided upper

H0: Variance of walk_speed <= Variance of run_speed

H1: Variance of walk_speed > Variance of run_speed

var.test(subset_3_long$speed ~ subset_3_long$movement, 
         alternative = "greater")

## 
##  F test to compare two variances
## 
## data:  subset_3_long$speed by subset_3_long$movement
## F = 0.2979, num df = 71, denom df = 71, p-value = 1
## alternative hypothesis: true ratio of variances is greater than 1
## 95 percent confidence interval:
##  0.2010832       Inf
## sample estimates:
## ratio of variances 
##          0.2979012

1-sided lower

H0: Variance of walk_speed >= Variance of run_speed

H1: Variance of walk_speed < Variance of run_speed

var.test(subset_3_long$speed ~ subset_3_long$movement, 
         alternative = "less")

## 
##  F test to compare two variances
## 
## data:  subset_3_long$speed by subset_3_long$movement
## F = 0.2979, num df = 71, denom df = 71, p-value = 3.888e-07
## alternative hypothesis: true ratio of variances is less than 1
## 95 percent confidence interval:
##  0.0000000 0.4413354
## sample estimates:
## ratio of variances 
##          0.2979012

5. ANOVA

Compare 3 groups for simplicity.

subset_4 <- subset(smash, game == "Pokemon" | game == "Super Mario" | 
                     game == "Fire Emblem")
head(subset_4)

##     character        game regular_fall fast_fall weight walk_speed run_speed
## 1       mario Super Mario         1.50     2.400     98      1.155     1.760
## 6       yoshi Super Mario         1.29     2.064    104      1.208     2.046
## 9     pikachu     Pokemon         1.55     2.480     79      1.302     2.039
## 10      luigi Super Mario         1.32     2.112     97      1.134     1.650
## 13 jigglypuff     Pokemon         0.98     1.568     68      0.735     1.271
## 14      peach Super Mario         1.19     1.904     89      0.924     1.595
##    full_hop_height short_hop_height double_hop_height before after expertise
## 1            36.33            17.54             36.33      3     8   Amateur
## 6            36.09            14.43             51.56      5     9   Amateur
## 9            35.50            17.12             35.50      8     9    Expert
## 10           44.00            19.98             41.31      7    10    Expert
## 13           19.79            11.26             19.79      3     7   Amateur
## 14           30.03            14.50             30.03      2     5   Amateur

H0: The means of fast_fall by game are equal

H1: The means of fast_fall by game are not equal

anova <- aov(fast_fall ~ game, data = subset_4)

summary(anova)

##             Df Sum Sq Mean Sq F value Pr(>F)
## game         2  0.353  0.1765    1.44   0.26
## Residuals   20  2.452  0.1226

Use post hoc test to determine the mean differences.

TukeyHSD((anova))

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = fast_fall ~ game, data = subset_4)
## 
## $game
##                               diff        lwr       upr     p adj
## Pokemon-Fire Emblem     -0.1624762 -0.6552818 0.3303294 0.6867043
## Super Mario-Fire Emblem -0.3051333 -0.7625510 0.1522844 0.2343060
## Super Mario-Pokemon     -0.1426571 -0.5791770 0.2938628 0.6911360

Test for normality.

shapiro.test(subset_4$fast_fall)

## 
##  Shapiro-Wilk normality test
## 
## data:  subset_4$fast_fall
## W = 0.91682, p-value = 0.05698

5.1 Two-Way ANOVA

When there is more than 1 variable to categorise the data, we can use a two-way ANOVA.

Use game and expertise.

anova_2 <- aov(regular_fall ~ game + expertise, 
               data = subset_4)
summary(anova_2)

##             Df Sum Sq Mean Sq F value Pr(>F)
## game         2 0.1162 0.05812   1.005  0.385
## expertise    1 0.1242 0.12418   2.147  0.159
## Residuals   19 1.0990 0.05784

In this case, the addition of expertise did not improve the model because of the lower residual sum of squares.

Use a post hoc test to determine the mean differences.

TukeyHSD(anova_2)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = regular_fall ~ game + expertise, data = subset_4)
## 
## $game
##                                diff        lwr       upr     p adj
## Pokemon-Fire Emblem     -0.05333333 -0.3932506 0.2865840 0.9165281
## Super Mario-Fire Emblem -0.16633333 -0.4818415 0.1491748 0.3916202
## Super Mario-Pokemon     -0.11300000 -0.4140937 0.1880937 0.6141903
## 
## $expertise
##                    diff         lwr       upr    p adj
## Expert-Amateur 0.140846 -0.06927372 0.3509656 0.176753

5.2 ANOVA with interaction variables

anova_3 <- aov(regular_fall ~ game * expertise, 
               data = subset_4)
summary(anova_3)

##                Df Sum Sq Mean Sq F value Pr(>F)
## game            2 0.1162 0.05812   1.002  0.388
## expertise       1 0.1242 0.12418   2.141  0.162
## game:expertise  2 0.1130 0.05650   0.974  0.398
## Residuals      17 0.9860 0.05800

TukeyHSD(anova_3)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = regular_fall ~ game * expertise, data = subset_4)
## 
## $game
##                                diff        lwr       upr     p adj
## Pokemon-Fire Emblem     -0.05333333 -0.3970504 0.2903838 0.9167871
## Super Mario-Fire Emblem -0.16633333 -0.4853684 0.1527018 0.3946154
## Super Mario-Pokemon     -0.11300000 -0.4174595 0.1914595 0.6158672
## 
## $expertise
##                    diff         lwr       upr     p adj
## Expert-Amateur 0.140846 -0.07124772 0.3529396 0.1791826
## 
## $`game:expertise`
##                                            diff        lwr       upr     p adj
## Pokemon:Amateur-Fire Emblem:Amateur     -0.3025 -0.9696311 0.3646311 0.6978071
## Super Mario:Amateur-Fire Emblem:Amateur -0.1605 -0.6772575 0.3562575 0.9137858
## Fire Emblem:Expert-Fire Emblem:Amateur   0.0925 -0.5746311 0.7596311 0.9974338
## Pokemon:Expert-Fire Emblem:Amateur       0.0895 -0.4272575 0.6062575 0.9927677
## Super Mario:Expert-Fire Emblem:Amateur  -0.1105 -0.6272575 0.4062575 0.9813904
## Super Mario:Amateur-Pokemon:Amateur      0.1420 -0.5025098 0.7865098 0.9788009
## Fire Emblem:Expert-Pokemon:Amateur       0.3950 -0.3753366 1.1653366 0.5853137
## Pokemon:Expert-Pokemon:Amateur           0.3920 -0.2525098 1.0365098 0.4102376
## Super Mario:Expert-Pokemon:Amateur       0.1920 -0.4525098 0.8365098 0.9264976
## Fire Emblem:Expert-Super Mario:Amateur   0.2530 -0.3915098 0.8975098 0.8038794
## Pokemon:Expert-Super Mario:Amateur       0.2500 -0.2372036 0.7372036 0.5846068
## Super Mario:Expert-Super Mario:Amateur   0.0500 -0.4372036 0.5372036 0.9993932
## Pokemon:Expert-Fire Emblem:Expert       -0.0030 -0.6475098 0.6415098 1.0000000
## Super Mario:Expert-Fire Emblem:Expert   -0.2030 -0.8475098 0.4415098 0.9091189
## Super Mario:Expert-Pokemon:Expert       -0.2000 -0.6872036 0.2872036 0.7742353

6. Kruskal-Wallis

For small samples, a Kruskal-Wallis can be used to determine mean differences.

kw <- kruskal.test(fast_fall ~ game, data = subset_4)
kw

## 
##  Kruskal-Wallis rank sum test
## 
## data:  fast_fall by game
## Kruskal-Wallis chi-squared = 3.0472, df = 2, p-value = 0.2179

Use a post hoc test to determine the mean differences. Or just do this to get the Kruskal-Wallis statistic in 1 step.

library(dunn.test)

dunn.test(subset_4$fast_fall, subset_4$game, kw = TRUE,
          table = TRUE, alpha = 0.05)

##   Kruskal-Wallis rank sum test
## 
## data: x and group
## Kruskal-Wallis chi-squared = 3.0472, df = 2, p-value = 0.22
## 
## 
##                            Comparison of x by group                            
##                                 (No adjustment)                                
## Col Mean-|
## Row Mean |   Fire Emb    Pokemon
## ---------+----------------------
##  Pokemon |   0.658023
##          |     0.2553
##          |
## Super Ma |   1.698708   1.037161
##          |     0.0447     0.1498
## 
## alpha = 0.05
## Reject Ho if p <= alpha/2

Next game!!

Super Inference Smash

master yeoda

a long, long time ago in a galaxy far, far away

Directions

1. Independent samples t-test

1.1 Load data

1.2 Independent samples t-test

1.3 Assumptions

2. One direction t-test

2.1 Subset data

2.2 Execute t-test

2.2.1 Less than alternate hypothesis

2.2.2 Greater than alternate hypothesis

2.3 Assumptions

3. Paired-samples t-test

3.1 Paired samples t-test

3.1 Less than alternate hypothesis

3.2 Greater than alternate hypothesis

3.4 Assumptions

4. F-Test

4.1 Alternate data format

5. ANOVA

5.1 Two-Way ANOVA

5.2 ANOVA with interaction variables

6. Kruskal-Wallis