Measurement

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<div class="my-footer">
<span>
<a href="https://psychmethods.github.io/coursenotes/" target="_blank">Methods in Psychological Research</a>
</span>
</div>

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# Measurement: The Foundation of Scientific Research

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## Roadmap

- Measurement
- Levels of measurement
- Goals of measurement

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## Understanding Measurement

- Definition: Measurement is the assignment of numbers to characteristics of people or objects
  - 4th step in the research process
  - It's purpose is to describe and differentiate 
--

- Examples: measurement scales
    - Speed `$\rightarrow$` miles-per-hour
    - Temperature `$\rightarrow$` Kelvin (K)
    - Order of finishers in a race `$\rightarrow$` 1st, 2nd, 3rd, etc.
    - Numbers on the back of basketball jerseys `$\rightarrow$` 28, 156, 152, 156, 215
    
---

## Variables: The Building Blocks of Measurement

- Variables contain the outcome of measurement processes
--

- Variables can be:
    - Qualitative: 
      - Numbers represent qualities (not quantities)
      - Example: Coding for eye color (1 = Blue, 2 = Brown, 3 = Green)
--
    - Quantitative: 
      - Numbers mean something in relation quantities in the real world
      - Example: Height in centimeters, Weight in kilograms
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# Levels of Measurement

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## Stevens' Classification (1946): 
.pull-left[
- Characterizes each level of measurement   
    - Nominal
    - ordinal, 
    - interval, 
    - ratio
]
--

.footnote[Stevens, Stanley Smith (1946). On the theory of scales of measurement. Science, 103(2684), 677-680.]

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## Key Properties of Measurement

- Have absolute zero
  - 0 indicated absence (origin means zero)
- Equal intervals
  - An interval means the same value at any point on measurement scales
- Order
  - Number means order
- Identity
  - Different numbers mean different measurement outcomes `$(1 \neq 2)$`

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## Which properties does this measure have?

<img src="data:image/png;base64,#../img/equalinterval.jpeg" width="65%" style="display: block; margin: auto;" />
--

.footnote[Source: [/u/ZigeonVO](https://www.reddit.com/user/ZigeonVO) via [r/assholedesign](https://www.reddit.com/r/assholedesign/comments/1fc7cnt/this_card_i_was_given_today_from_a_delivery/)]
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# Ratio (quantitative)

- All four properties
    - Have absolute zero
    - Equal intervals
    - Order
    - Identity
--
- Multiplication and division are permissible transformations
--

- Examples: 
    - Height, weight, age, income, time, etc.

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# Example (Base R)

``` r
# loads the HistData package
library(HistData)
# loads the Galton dataset
data("Galton")
# First 3n rows of data
head(Galton, n=3)
```

```
##   parent child
## 1   70.5  61.7
## 2   68.5  61.7
## 3   65.5  61.7
```

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# Example (Base R)

.pull-left[
<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-5-1.png" width="100%" style="display: block; margin: auto;" />

]
.pull-right[

``` r
# Histogram
hist(Galton$child)
```
]

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# Example (Base R)

.pull-left[
<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-6-1.png" width="100%" style="display: block; margin: auto;" />
]
.pull-right[

``` r
# Density Plot
plot(density(Galton$child))
```

]

---

# Bandwidth Aside

[Bandwidth: Smoothing Method](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/density.html)

``` r
args(density.default)
```

```
## function (x, bw = "nrd0", adjust = 1, kernel = c("gaussian", 
##     "epanechnikov", "rectangular", "triangular", "biweight", 
##     "cosine", "optcosine"), weights = NULL, window = kernel, 
##     width, give.Rkern = FALSE, subdensity = FALSE, warnWbw = var(weights) > 
##         0, n = 512, from, to, cut = 3, ext = 4, old.coords = FALSE, 
##     na.rm = FALSE, ...) 
## NULL
```

``` r
#?density # Gives you documentation
```

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# Bandwidth Aside
<img src="data:image/png;base64,#../img/density.png" width="90%" style="display: block; margin: auto;" />

---

# Bandwidth Aside

.pull-left[
<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-9-1.png" width="100%" style="display: block; margin: auto;" />
]
.pull-right[

``` r
set.seed(201010)
x <- rnorm(1000, 10, 2)
par(mfrow = c(2,2))

#A bit bumpy
plot(density(x))
#Very sooth
plot(density(x,adjust = 10))
 #Very bumpy
plot(density(x,adjust = .1))
```

]

---

# Interval (quantitative)

- Has order, identity, and equal intervals
   - (all but absolute zero)
- Addition is a permissible transformation
- Example: Temperature in Celsius or Fahrenheit

``` r
# Interval Example
library(datasets)
data("nottem")
nottem[1:10] 			# First ten rows of data
```

```
##  [1] 40.6 40.8 44.4 46.7 54.1 58.5 57.7 56.4 54.3 50.5
```

---

## Interval (quantitative)

``` r
# Histogram
hist(nottem)
```

<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-11-1.png" width="90%" style="display: block; margin: auto;" />
]
.pull-right[

``` r
# Density Plot
plot(density(nottem))
```

<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-12-1.png" width="90%" style="display: block; margin: auto;" />
]

---

# Ordinal (qualitative)

.pull-left-narrow[
-	Has order and identity
- Monotonic transformations are permissible
- These variables maintain the order of the values
- Example: Education levels (High School, Bachelor's, Master's, PhD)
]
--
.pull-right-wide[

``` r
# Ordinal Example
library(ggplot2movies)
data(movies)
# First 4 rows of data, with a non-missing budget
head(movies[!is.na(movies$budget),], n = 14) 
```

```
## # A tibble: 14 × 24
##   title   year length budget rating votes    r1    r2    r3    r4
##   <chr>  <int>  <int>  <int>  <dbl> <int> <dbl> <dbl> <dbl> <dbl>
## 1 'G' M…  1935     85 4.50e5    7.2   281   0     4.5   4.5   4.5
## 2 'Mano…  1966     74 1.9 e4    1.6  7996  74.5   4.5   4.5   4.5
## 3 'Til …  1997    113 2.3 e7    4.8   799   4.5   4.5   4.5  14.5
## 4 .com …  2002     96 5   e6    3.7   271  64.5   4.5   4.5   4.5
## 5 10 Th…  1999     97 1.6 e7    6.7 19095   4.5   4.5   4.5   4.5
## 6 100 M…  2002     98 1.10e6    5.6   181   4.5   4.5   4.5   4.5
## # ℹ 8 more rows
## # ℹ 14 more variables: r5 <dbl>, r6 <dbl>, r7 <dbl>, r8 <dbl>,
## #   r9 <dbl>, r10 <dbl>, mpaa <chr>, Action <int>,
## #   Animation <int>, Comedy <int>, Drama <int>,
## #   Documentary <int>, Romance <int>, Short <int>
```
]
---

# Ordinal (qualitative)

``` r
# Histogram
variable <- movies$rating
hist(variable) 
```

<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-14-1.png" width="90%" style="display: block; margin: auto;" />
]
.pull-right[

``` r
# Density Plot
plot(density(variable)) 
```

<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-15-1.png" width="90%" style="display: block; margin: auto;" />
]

---

# Nominal

.pull-left-narrow[
- Only has identity
- Any identity-preserving transformation is permissible
- Example: Jersey numbers in sports
]

``` r
# Nominal Example
library(vcd)
data(Arthritis)
head(Arthritis[, c("ID", "Treatment")], 8)
```

```
##   ID Treatment
## 1 57   Treated
## 2 46   Treated
## 3 77   Treated
## 4 17   Treated
## 5 36   Treated
## 6 23   Treated
## 7 75   Treated
## 8 39   Treated
```

---

# Basic Barplot
.pull-left[
<img src="data:image/png;base64,#measurement_files/figure-html/unnamed-chunk-17-1.png" width="100%" style="display: block; margin: auto;" />
]
.pull-right[

``` r
variable<-Arthritis$Treatment
#hist(variable) # error

barplot_fix <-
     prop.table(table(variable))
# Sometimes, R is silly
barplot(barplot_fix)
```
]
---

# More complex measurement level taxomony

- Missing (considered nominal under the Stevens)
- Binary (considered nominal under the Stevens)
- Nominal (considered nominal under the Stevens)
- Partially ordered (considered ordinal under Stevens)
- Fully ordered (considered ordinal under Stevens)
- Interval
- Ratio
- Absolute measurement (has no permissible transformation)
    - `$6.02$` x `$10^{23}$`
    - `$\pi$`
    
---

## Could measurement level be itself on a continuum?

.pull-left[
- Example: IQ
    - Falls between interval and ratio?
    - Or falls between ordinal and interval?
![](data:image/png;base64,#../img/iq_curves.gif)
]

--
.pull-right[
 ![](data:image/png;base64,#../img/iq_fail.png)
]

---

# Goals of Measurement

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# Goals of Measurement

- Reliability
  - "…the degree to which a test or measure produces the same scores when applied in the same circumstances…" (Thomas and Nelson 1996)
  - In other words, if you take the measure again, will you get the same result?

--
- (Internal) Validity
  - "Degree to which a test or instrument measures what it purports to measure"  (Thomas and Nelson 1996)
  - In other words, does your measure measure what is it supposed to measure?

---

## Types of Validity (More on this later...)

- Two Major Areas within Validity
  - **Internal Validity**
      - Is this evidence supportive of our claim, within this study?
  - **External Validity**
      - Is this evidence supportive of our claim beyond this study?
      - Does this finding generalize to outside this study?

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## Practical Applications

- Psychology: Measuring personality traits
- Education: Assessing student performance
- Medicine: Tracking patient vital signs
- Business: Evaluating customer satisfaction
- Sports: Analyzing athletic performance

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# Challenges in Measurement

- Measurement Error
  - Random errors vs. Systematic errors
- Bias in Measurement
  - Observer bias, response bias, selection bias
- Cultural and Linguistic Considerations
  - Ensuring measures are valid across different cultures and languages
- Ethical Considerations
  - Privacy, consent, and responsible use of data

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## The Importance of Good Measurement

<br><br>
![](data:image/png;base64,#../img/validreliable.gif)

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# Lingering Questions
- How might the level of measurement affect the statistical analyses we can perform?
- Can you think of examples where improper measurement could lead to incorrect conclusions?
- How can we improve the reliability and validity of our measurements in real-world research?

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# Wrapping Up...