Center and Spread

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# Center and Spread
## Cohen Chapter 3 .small[EDUC/PSY 6600]

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## "You can, for example, never foretell what any one man will do, but you can say with precision what an average number will be up to. *Individuals vary*, but percentages remain constant. So says the statistician." 
### -- Sherlock Holmes, *The Sign of Four*

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# Distributions Examples

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# Three Measures of Center

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# Mean vs. Median

.large[.large[
.nicegreen[Median]: the center point, half of values are on each side, not affected by the skew, the "typical value"

.dcoral[Mean]: the "balance" point, pulled to the side of the skew, not typical

]]

.large[If distribution is symmetrical: mean = median]

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# Distributions and Numbers

.pull-left[
.large[
- The MEDIAN is **resistant** & doesn't change much
- The MEAN is **influenced** & changes more!
- Average does NOT mean typical
- Average moves when we remove the high point
]]

.pull-right[
<img src="u01_Ch3_CenterSpread_files/figure-html/unnamed-chunk-1-1.png" style="display: block; margin: auto;" />
]

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# Distributions and Numbers

.pull-left[
.large[
- The MEDIAN is **resistant** & doesn't change much
- The MEAN is **influenced** & changes more!
- Average does NOT mean typical
- Average moves when we remove the high point
- Median doesn't move when we remove the high point
]]

.pull-right[
<img src="u01_Ch3_CenterSpread_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" />
]

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# Three Measures of Spread

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# Best Summary of the Data?

.huge[
"... the perfect estimator does not exist." -- Rand Wilcox, 2001
]

.pull-left[
.large[
## .bluer[Median and SIR]

Skewed data or outliers
]]

.pull-right[
.large[
## .nicegreen[Mean and SD]

Symmetrical and no outliers
]]

.large[.large[
A .dcoral[graph gives the best overall picture of a distribution]
]]

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# Properties of the Mean and SD

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# Skewness

.pull-left[
.large[
- Degree of .dcoral[symmetry] in distribution
- Can detect **visually** (histogram, boxplot)
- Skewness statistic
    - Based on cubed deviations from the mean
    - Divided by SE of skewness
    - `$> \pm 2$` is a sign of skewed data

]]

.pull-right[
.large[
$$
Skewness = \frac{N}{N - 2}\frac{\sum_{i=1}^n (X_i - \bar{X})^3}{(N - 1)s^3}
$$

- Interpreting skewness statistic
    - positive value = positive (right) skew
    - negative value = negative (left) skew
    - zero value = no skew
]]

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# Kurtosis

$$
Kurtosis = \frac{N(N+1)}{(N - 2)(N - 3)}\frac{\sum_{i=1}^n (X_i - \bar{X})^4}{(N - 1)s^4} - 3 \frac{(N - 1)(N - 1)}{(N - 2)(N - 3)}
$$

.pull-left[
.large[
- Degree of .dcoral[flatness] in distribution
- Harder to detect visually
- Kurtosis statistic
    - Based on deviations from the mean (raised to 4th power)
    - Divided by SE of kurtosis
    - `$> \pm 2$` is a sign of problems with kurtosis
]]

.pull-right[
.large[
- Interpreting kurtosis statistic
    - positive value = leptokurtic (peaked)
    - negative value = platykurtic (flat)
    - zero value = mesokurtic (normal)
]]

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# Kurtosis

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# Five-Number Summary

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# Five-Number Summary - Median

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# Five-Number Summary - Quartiles

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# Boxplots (Modified) - Lines

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# Boxplots (Modified) - IQR and SIQR

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# Boxplot vs. Histogram

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# Boxplots by Group

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# Density Plots

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# Quantile-Quantile (Q-Q) Plot

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# Let's Apply This To the Cancer Dataset (on Canvas)

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# Read in the Data

```r
library(tidyverse)    # Loads several very helpful 'tidy' packages
library(rio)          # Read in SPSS datasets
library(furniture)    # Nice tables (by our own Tyson Barrett)
library(psych)        # Lots of nice tid-bits
```

```r
cancer_raw <- rio::import("cancer.sav")
```

--
### And Clean It

```r
cancer_clean <- cancer_raw %>% 
 dplyr::rename_all(tolower) %>% 
 dplyr::mutate(id = factor(id)) %>% 
 dplyr::mutate(trt = factor(trt,
 labels = c("Placebo", 
 "Aloe Juice"))) %>% 
 dplyr::mutate(stage = factor(stage))
```

---
## Frequency Tables with `furniture::tableF()`

.pull-left[

```r
cancer_clean %>%
  furniture::tableF(age, n = 8)
```

```

──────────────────────────────────
 age Freq CumFreq Percent CumPerc
 27  1    1       4.00%   4.00%  
 42  1    2       4.00%   8.00%  
 44  1    3       4.00%   12.00% 
 46  2    5       8.00%   20.00% 
 ... ...  ...     ...     ...    
 68  1    20      4.00%   80.00% 
 69  1    21      4.00%   84.00% 
 73  1    22      4.00%   88.00% 
 77  2    24      8.00%   96.00% 
 86  1    25      4.00%   100.00%
──────────────────────────────────
```
]

.pull-right[

```r
cancer_clean %>%
  furniture::tableF(trt)
```

```

─────────────────────────────────────────
 trt        Freq CumFreq Percent CumPerc
 Placebo    14   14      56.00%  56.00% 
 Aloe Juice 11   25      44.00%  100.00%
─────────────────────────────────────────
```
]

---
## Extensive Descriptive Stats `psych:describe()`

```r
cancer_clean %>% 
  dplyr::select(age, weighin, totalcin, totalcw2, totalcw4, totalcw6) %>%
  psych::describe()
```

```
         vars  n   mean    sd median trimmed   mad min   max range  skew
age         1 25  59.64 12.93   60.0   59.95 11.86  27  86.0  59.0 -0.31
weighin     2 25 178.28 31.98  172.8  176.57 21.05 124 261.4 137.4  0.73
totalcin    3 25   6.52  1.53    6.0    6.33  0.00   4  12.0   8.0  1.80
totalcw2    4 25   8.28  2.54    8.0    8.10  2.97   4  16.0  12.0  1.01
totalcw4    5 25  10.36  3.47   10.0   10.19  2.97   6  17.0  11.0  0.49
totalcw6    6 23   9.48  3.49    9.0    9.21  2.97   3  19.0  16.0  0.77
         kurtosis   se
age         -0.01 2.59
weighin      0.07 6.40
totalcin     4.30 0.31
totalcw2     1.14 0.51
totalcw4    -1.00 0.69
totalcw6     0.53 0.73
```

---
## Smaller Set with `furniture::table1()`

.pull-left[
For the Entire Sample

```r
cancer_clean %>%
  furniture::table1(trt, age, weighin)
```

```

─────────────────────────────────
               Mean/Count (SD/%)
               n = 25           
 trt                            
    Placebo    14 (56%)         
    Aloe Juice 11 (44%)         
 age                            
               59.6 (12.9)      
 weighin                        
               178.3 (32.0)     
─────────────────────────────────
```
]

.pull-right[
Breaking the Sample by a Factor

```r
cancer_clean %>%
  dplyr::group_by(trt) %>%        
  furniture::table1(age, weighin)
```

```

───────────────────────────────────
                  trt 
         Placebo      Aloe Juice  
         n = 14       n = 11      
 age                              
         59.8 (9.0)   59.5 (17.2) 
 weighin                          
         167.5 (23.0) 192.0 (37.4)
───────────────────────────────────
```
]

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## Boxplot, one one `geom_boxplot()`

```r
cancer_clean %>%
  ggplot(aes(x = "Full Sample",   # x = "quoted text"
             y = age)) +          # y = contin_var (no quotes)
  geom_boxplot()
```

---
## Boxplots, by groups - (1) fill color

```r
cancer_clean %>%
  ggplot(aes(x = "Full Sample",           # x = "quoted text"
             y = age,                     # y = contin_var (no quotes)
             fill = trt)) +               # fill = group_var (no quotes) 
  geom_boxplot()
```

---
## Boxplots, by groups - (2) x-axis breaks

```r
cancer_clean %>%
  ggplot(aes(x = trt,             # x = group_var (no quotes)  
             y = age)) +          # y = contin_var (no quotes)
  geom_boxplot()
```

---
## Boxplots, by groups - (3) seperate panels

```r
cancer_clean %>%
  ggplot(aes(x = "Full Sample",   # x = "quoted text"
             y = age)) +          # y = contin_var (no quotes)
  geom_boxplot() +
  facet_grid(. ~ trt)             # . ~ group_var (no quotes)
```

---
## Boxplot for a Subset - 1 requirement

```r
cancer_clean %>% # Less than 172 Pound at baseline
 dplyr::filter(weighin < 172) %>%
 ggplot(aes(x = "Weigh At Baseline < 172", 
 y = age)) +
 geom_boxplot()
```

---
## Boxplot for a Subset - 2 requirements

```r
cancer_clean %>%           # At least 150 pounds AND not in Aloe group
  dplyr::filter(weighin >= 150 & trt == "Placebo") %>%
  ggplot(aes(x = "Placebo and at least 150 Pounds", 
             y = age)) +
  geom_boxplot()
```

---
## Boxplot for a Subset - 2 requirements (`%in%`)

```r
cancer_clean %>%          # In Aloe group, but only stages 2-4
  dplyr::filter(trt == "Aloe Juice" & stage %in% c(2, 3, 4)) %>%
  ggplot(aes(x = "On Aloe Juice and Stage 2-4", 
             y = weighin)) +
  geom_boxplot()
```

---
## Boxplot for Repeated Measures

```r
cancer_clean %>%
  tidyr::gather(key = "time",         # stack the repeated measures
                value = "value", 
                totalcin, totalcw2, totalcw4, totalcw6) %>%
  ggplot(aes(x = time, 
             y = value)) +
  geom_boxplot()
```

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## Boxplot: COMPLICATED!

```r
cancer_clean %>%
  dplyr::filter(weighin > 130 & stage %in% c(2, 4)) %>%
  tidyr::gather(key = "time", value = "value",
                totalcin, totalcw2, totalcw4, totalcw6) %>%
  ggplot(aes(x = time, y = value, fill = stage)) +
  geom_boxplot() +
  facet_grid(. ~ trt)
```

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# Questions?

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# Next Topic

### Standard and Normal