An Introduction to [ comment ] and RStudio for Educational Researchers

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.title[
# An Introduction to <svg viewBox="0 0 581 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;position:relative;display:inline-block;top:.1em;fill:#384CB7;"> [ comment ] <path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z" /></svg>and RStudio for Educational Researchers
]
.subtitle[
## <br>Descriptive and Inferential Statistics:<br>Correlation
]
.author[
### Jorge Sinval
]
.date[
### 2025-11-18
]

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# 4. Correlation

---

# Pearson correlation

.panelset[
.panel[.panel-name[Test]

**Pearson correlation coefficient significance test**

Is the observed correlation statistically significant?

Can we assume that in the population the correlation is significantly different from zero?

]

.panel[.panel-name[Assumptions]

1. `$X_1$` e `$X_2$` quantitative

2. `$X_1$` e `$X_2$` linearly related

3. `$X_1$` e `$X_2$` with normal distribution (bivariate)

]

.panel[.panel-name[Hypotheses]

`$H_0: \rho = 0$` vs. `$H_1: \rho \neq 0$`

]

.panel[.panel-name[Test statistic]

$$T_R=\frac{R}{\sqrt{\frac{1-R^2}{n-2}}} \sim t(n-2) $$

]

.panel[.panel-name[Decision]

Reject `$H_0$` if `$|T_R| \geq t_{1−\frac{\alpha}{2};(n−2)}$`

On <svg viewBox="0 0 581 512" xmlns="http://www.w3.org/2000/svg" style="height:1em;position:relative;display:inline-block;top:.1em;fill:#384CB7;">  [ comment ]  <path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"></path></svg>: Reject `$H_0$` if `$p-value \leq \alpha$`

]

.panel[.panel-name[Conclusion]

The variables `$X_1$` and `$X_2$` [are/are not] significantly correlated `$(t_r(df) = t.ttt; r = .rrr; p = .ppp; N = n)$`. The correlation is [nule/small/moderate/large] [-/and positive/and negative].

]

.panel[.panel-name[R code]

Example:

_Investigate if work engagement and burnout are significantly correlated._

<div class="pre-name">pearson_cor_test.R</div>

``` r
ds <- readr::read_csv(trimws("https://ndownloader.figshare.com/files/22299075 "))
ds$we <- rowMeans(ds[,paste0("UWES",1:9)]) #create work engagement variable
ds$burnout <- rowMeans(ds[,paste0("OLBI",c(1:12,14:16))]) #create burnout variable
library(ggplot2)
ggplot(ds, aes(x = we, y = burnout)) + #scatter plot
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  labs(title = "Work engagement and burnout",
       x = "Work engagement",
       y = "Burnout") +
LittleHelpers::theme_mr() #package LittleHelpers to provide a APA ready theme
*cor.test(x = ds$we,y = ds$burnout, method = "pearson")
```
]

.panel[.panel-name[Output]

```
## 
## 	Pearson's product-moment correlation
## 
## data:  ds$we and ds$burnout
*## t = -33.7, df = 1080, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.7437948 -0.6855893
## sample estimates:
##        cor 
*## -0.7159338
```

]

---

# Spearman correlation

.panelset[
.panel[.panel-name[Test]

**Spearman correlation coefficient significance test**

Is the observed correlation statistically significant?

Can we assume that in the population the correlation is significantly different from zero?

]

.panel[.panel-name[Assumptions]

1. `$X_1$` and `$X_2$` measured on an at least ordinal measurement scale

]

.panel[.panel-name[Hypotheses]

`$H_0: \rho_s = 0$` vs. `$H_1: \rho_s \neq 0$`

]

.panel[.panel-name[Test statistic]

`$T_{R_S} =R_S \sqrt{\frac{n-2}{1-R_S^2}} \sim t(n-2)$`

]

.panel[.panel-name[Decision]

Reject `$H_0$` if `$|T_{R_S}| \geq t_{1−\frac{\alpha}{2};(n−2)}$`

]

.panel[.panel-name[Conclusion]

The variables `$X_1$` and `$X_2$` [are/are not] significantly correlated `$(t_{r_S}(df) = t.ttt; r = .rrr; p = .ppp; N = n)$`. The correlation is [nule/small/moderate/large] [-/and positive/and negative].

]

.panel[.panel-name[R code]

Example:

_Investigate if there is an association between OLBI's 8<sup>th</sup> item (i.e., "During my work, I often feel emotionally drained") and UWES' 1<sup>st</sup> item (i.e., "At my work, I feel bursting with energy").UWES' items are answered on a 7-point ordinal scale (0 &mdash; "Never" to 6 &mdash; "Always"), while OLBI's items are answered on a 5-point Likert scale (1 &mdash; "Strongly Disagree" to 5 &mdash; "Strongly Agree")_.

<div class="pre-name">spearman_cor_test.R</div>

``` r
ds <- readr::read_csv(trimws("https://ndownloader.figshare.com/files/22299075 "))
*cor.test(x = ds$OLBI8,y = ds$UWES1, method = "spearman", exact = T)
```
]

.panel[.panel-name[Output]

```
## 
## 	Spearman's rank correlation rho
## 
## data:  ds$OLBI8 and ds$UWES1
*## S = 290153241, p-value < 2.2e-16
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##        rho 
*## -0.3743497
```

]

---

# Cramér's V correlation

.panelset[
.panel[.panel-name[Test]

**Cramér's V correlation coefficient significance test**

Is the observed correlation statistically significant?

Can we assume that in the population the correlation is significantly different from zero?

]

.panel[.panel-name[Assumptions]

1. `$X_1$` and `$X_2$` are on (at least) nominal measurement scale with the data organized in a contingency table
2. Independent samples
3. `$N>20$`
4. At least `$80\% E_{ij} \geq5$`
5. `$100\% E_{ij} >1$`

]

.panel[.panel-name[Hypotheses]

`$H_0: \rho_v = 0$` vs. `$H_1: \rho_v \neq 0$`

]

.panel[.panel-name[Test statistic]

`$X^2= \sum\limits_{i=1}^R\sum\limits_{j=1}^C \frac{(O_{ij}-E_{ij})^2}{E_{ij}} \sim \chi^2_{(R-1)(C-1)}$`

]

.panel[.panel-name[Decision]

Reject `$H_0$` if `$X^2 \geq \chi^2_{1-\alpha;(R-1)(C-1)}$`

]

.panel[.panel-name[Conclusion]

The variables `$X_1$` and `$X_2$` [are/are not] significantly correlated `$(\chi^2_{(df)} = x.xxx; V = .vvv; p = .ppp; N = n)$`. The correlation is [nule/small/moderate/large].

]

.panel[.panel-name[R code]

Example:

_Investigate if sex and country are independent._

<div class="pre-name">cramer_v_test.R</div>

``` r
ds <- readr::read_csv(trimws("https://ndownloader.figshare.com/files/22299075 "))
*library(lsr)
contigency_table <- table(ds$Sex, ds$Country) #contingency table with qualitative variables
*chisq.test(contigency_table,correct = F) #significance test
*cramersV(contigency_table) #Cramér V estimate
```

.pull-left[`correct` argument can be:
  * `TRUE` (default) applies continuity correction when computing the test statistic for 2 by 2 tables; one half is subtracted from all `$|O-E|$` differences: `$X^2= \sum\limits_{i=1}^R\sum\limits_{j=1}^C \frac{(|O_{ij}-E_{ij}|-0.5)^2}{E_{ij}} \sim \chi^2_{(R-1)(C-1)}$`;  
  ]
.pull-right[
  * `FALSE` does not apply continuity correction: `$X^2= \sum\limits_{i=1}^R\sum\limits_{j=1}^C \frac{(O_{ij}-E_{ij})^2}{E_{ij}} \sim \chi^2_{(R-1)(C-1)}$`.  
  ]

]

.panel[.panel-name[Output]

```
## 
## 	Pearson's Chi-squared test
## 
## data:  contigency_table
*## X-squared = 2.2127, df = 1, p-value = 0.1369
## 
*## [1] 0.0440688
```

]

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# More info

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Practice is the best strategy for learning.

_In God we trust, all others bring data_

Edwards Deming

THE END

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