Descriptive Statistics

# Biostatistics

## Descriptive Statistics

Michael Luu, MPH

Biostatistics and Bioinformatics Research Center | Cedars Sinai Medical Center

#### June 22, 2021

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# Introduction

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# What is descriptive statistics?

- **Descriptive statistics** is a collection of statistical measures and tools used to give us a better sense of the sampled data

- Not to be confused with **inferential statistics** where we are trying to reach
conclusions that extend beyond the sampled data

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# Why descriptive statistics ?

- Provides an understanding of the underlying sample population

- Simplifies large amounts of data to a simpler summary

- Identifies potential measurement errors or mistakes

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# Types of Data

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# Types of Data

### Quantitative (Numeric)

Variable that has been measured on a numeric or quantitative scale

#### **Continuous** 
  
- Can theoretically take on an infinite number of values - accuracy is limited only by the measuring instrument 
  - e.g. age, BMI, BSA, height, weight, etc..

#### **Discrete** 
- Numerical variables that are measured and can only be whole numbers 
  - e.g. age, heart rate, number of medication taken, number of relapses, etc..

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# Types of Data

### Qualitative (Categorical)

Variables that are typically not directly measured by an instrument, and are based on observations

#### **Ordinal** 
- Variables that have an inherent hierarchical order to the relationship among the different categories 
  - e.g. pain scores, stage of cancer, education level, etc..

#### **Nominal** 
- Variables that are "named" or classified into one or more qualitative groups
- Do not have a sense of ordering between the different categories
  - e.g. risk factors, types of medications consumed, types of symptoms experienced, surgical outcomes, blood type, gender, etc..

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# Types of Data

### Why do we need to identify the types of data ?

- In statistics we have specialized tools or measures to handle different type of data

- You will **NEED** to understand what kind of data you have in order to correctly summarize your data

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# Quantitative Data

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# Numerical Summarization

.pull-right[
### Measures of Variability or Dispersion
  - Minimum and Maximum
  
  - Percentiles / Interquartile Range (IQR)
  
  - Standard Deviation

]

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# Measures of Location

### Mean

- The sample mean is the most commonly used and readily understood measure of central tendency.

- The sample mean can be defined as:

`$$\huge \bar{x} = \frac{\sum x_i}{n}$$`

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# Example Data

| patient_id|sex | age|
|----------:|:---|---:|
|          1|M   |  32|
|          2|F   |  35|
|          3|F   |  35|
|          4|F   |  34|
|          5|M   |  36|
|          6|F   |  36|
|          7|F   |  30|
|          8|F   |  36|
|          9|F   |  33|
|         10|F   |  32|

---

# Example

We have a collection of **age** that was collected from the sample population

```
 [1] 32 35 35 34 36 36 30 36 33 32
```

* The **total sum** of all of the sampled age is 339

* The **total number of measurements** collected is 10

* The **mean** or average age among the sample population is 33.9

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# Measures of Location
### Median

- The median is the midpoint of the values

--
  -  The midpoint value is the point at which half the observations are above the value and half the observations are below the value *(50th percentile*). 
  
--

- If there are two 'middle' values then the median is the average of the two mid values

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# Example

Let's recall the collection of **age** that we have collected from the sample population

```
 [1] 32 35 35 34 36 36 30 36 33 32
```

* We start by ranking the data from smallest to largest

```
 [1] 30 32 32 33 34 35 35 36 36 36
```

* We identify the **middle** value from the data

```
[1] 34 35
```

* We then take the average of the two middle value to obtain the **median**

```
[1] 34.5
```

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# Measures of Location
### Mode

- The mode is the value that appears most often in a set of values.

- Not always a measure of central tendency

- The mode is only useful for discrete values or continuous values with limited digits of accuracy

- It's possible to have more than 1 mode

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# Example

```
 [1] 32 35 35 34 36 36 30 36 33 32
```

- Let's tabulate the occurences of each of the sampled age

```

30 32 33 34 35 36 
 1  2  1  1  2  3 
```

- The mode is the value that occured the most often.

- In our example, the mode is **36**

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# Measures of Variability or Dispersion

### Minimum and Maximum

A measure of dispersion and is defined as the smallest and largest value

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# Example

```
 [1] 32 35 35 34 36 36 30 36 33 32
```

- We will rank the data from smallest to largest

```
 [1] 30 32 32 33 34 35 35 36 36 36
```

- The min and max corresponds to the smallest and largest values of our sample.

- In our example, the min would be **30**, and the max would be **36**

---
# Measures of Variability or Dispersion

### Percentiles / Interquartile Range (IQR)

- The interquartile range is defined as the range between the 25th and 75th percentiles

`$$\LARGE IQR = Q_3 - Q_1$$`

- It is commonly denoted after presenting the median

- The interquartile range can be used to describe the spread of the data. As the spread of the data increases, the IQR becomes larger.

- It is also used to build box plots.

- Depending on the statistical software you are using, there are multiple ways of calculating quantile

---
# Example

We determine the interquartile range by calculating quantiles at the 25th and the 75th percentile.

```
 [1] 32 35 35 34 36 36 30 36 33 32
```

- We begin by sorting the values

```
 [1] 30 32 32 33 34 35 35 36 36 36
```

- The 25th percentile is the value that divides the data where 25 percent falls below this value and 75 percent falls above this value.

- The 75th percentile is the value that divides the data where 75 percent falls below this value and 25 percent falls above this value.

```
  0%  25%  50%  75% 100% 
30.0 32.0 34.5 36.0 36.0 
```

- The interquantile range is the distance between the value of the 75th percentile and the 25th percentile.

- In this example, the IQR is **4**

---

# Measures of Variability or Dispersion
### Standard Deviation

The standard deviation is a measure of how spread out the data are about the mean.

`$$\LARGE s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2}$$`

---

# Example

```
 [1] 32 35 35 34 36 36 30 36 33 32
```

- The mean sample age is **33.9**

- There is a total of **10** measurements

]

```
# A tibble: 10 × 4
     age mean_age   diff   diff2
   <int>    <dbl>  <dbl>   <dbl>
 1    32     33.9 -1.90   3.61  
 2    35     33.9  1.10   1.21  
 3    35     33.9  1.10   1.21  
 4    34     33.9  0.100  0.0100
 5    36     33.9  2.10   4.41  
 6    36     33.9  2.10   4.41  
 7    30     33.9 -3.9   15.2   
 8    36     33.9  2.10   4.41  
 9    33     33.9 -0.900  0.810 
10    32     33.9 -1.90   3.61  
```

]

---
# Example

```
# A tibble: 1 × 4
  summ_diff2 total_length_minus_one   var    sd
       <dbl>                  <dbl> <dbl> <dbl>
1       38.9                      9  4.32  2.08
```

The sample standard deviation is **2.08**

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# Example

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# Presenting / Describing your data

- You should always include **BOTH** a *measure of central tendency* and *measure of dispersion* when presenting your data

- The choice is dependent on the distribution of your data

- mean ± SD
- median (25%, 75% quantiles)
- median (IQR)
- median (min - max)

]

- median (25%, 75% quantiles)
- median (IQR)
- median (min - max)

]
---

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# Summarizing Qualitative Data

- The primary method of summarizing qualitative data is frequency **counts** and **percentages**

```
 [1] "M" "F" "F" "F" "M" "F" "F" "F" "F" "F"
```

- Let's tabulate the total occurrences of M and F in our data

```
# A tibble: 2 × 2
  sex       n
  <chr> <int>
1 F         8
2 M         2
```

- Next we take the total and divide by the total number of patients

```
# A tibble: 2 × 3
  sex       n proportion
  <chr> <int> <chr>     
1 F         8 80.0%     
2 M         2 20.0%     
```

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# Graphical Summarizations

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# What do the following figures have in common?

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# They all have the same summary statistic ...

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https://www.autodesk.com/research/publications/same-stats-different-graphs

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# Quantitative summary measures are useful ...

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# Graphical summarizations provides an additional perspective

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# Graphical Summarization

## Quantitative Data

- Histograms
- Dot plots
- Box plots

## Qualitative Data

- Bar plots

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# Histogram

* Useful for all sized data (small and large)

* Allows us to visualize the spread and distribution of continuous variables

* Each bar represents a 'bin' or a defined interval of values

* Although not as common, the width of the bins does NOT have to be equal!

* The y axis or the height of the bar represents the count of the number of values that fall into each bin
  
* The y axis is also commonly normalized to 'relative' frequencies to show the proportion of cases or density that falls into each bin.
]

]

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# Distribution

> "A distribution is simply a collection of data, or scores, on a variable. Usually, these scores are arranged in order from smallest to largest and then they can be presented graphically."

— Page 6, Statistics in Plain English, Third Edition, 2010.

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# Example (n = 10, binwidth = 1)

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# Example (n = 500, binwidth = 1)

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# Example (n = 500, binwidth = 5)

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# Dot plot

* Useful for small to moderate sized data

* Allows us to visualize the spread and distribution of one continuous discrete variables
  * e.g. length of stay

* The X axis is the variable of interest and each dot represents a single observation

* Easy to identify the mode

* Highlights clusters, gaps, and outliers

* Intuitive and easy to understand

]

]

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# Example

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# Box plots

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# Example

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# Boxplots are not perfect ...

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# Raw Data vs Box Plot vs Violin Plot

https://www.autodesk.com/research/publications/same-stats-different-graphs

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# How are violin plots made?

* Hintze, J. L., & Nelson, R. D. (1998). Violin plots: a box plot-density trace synergism. The American Statistician, 52(2), 181-184.

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# Violin plot

* Violin plots are box plots, with an overlay of the density distribution (histogram) of the data

* More informative than a simple box plot

* Visualizes the full distribution of the data

* Especially useful for bimodal or multimodal distribution
  * e.g. distribution of data with multiple peaks

]

]

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# Example

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# Bar plot

* Useful for visualizing **categorical** data

* Commonly used to present counts and proportion of each level

* Allows us to quickly observe the difference in magnitude of each level based on the height of each bar

]

]

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# Bar plot

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# Bar plot

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# Summary

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# Summary

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# Questions