Frequencies

• Simply how often something occurs
• But can be expressed differently:
1. Relative to the entire sample
   • “Of all 100 patients, 80 had pain disorders”
   • Presented as percents, probabilities, or risks
2. Relative to other subgroups
   • “80 of these patients developed pain disorders, and 20 did not”
   • Presented as odds
   • “The odds of these patients developing a pain disorder were 4:1”

Relative Frequencies

• We’re often interested not only in how something happens
  • But if it’s more/less common in one group/situation than in others
• This is the relative frequency
  • Can be expressed as:
    • Relative risk (relative probability)
    • Relative odds
  • These are usually instead written as:
    • Risk ratio
    • Odds ratio

Risks/Probabilities

• Again, risk & probability are the same thing
• Expressed as the chance of something happening standardized to a scale of 0 to 1
  • Often computed as the chance—or number of times—something happens out of all possible times

\[P(\mathsf{Target\ outcome}) = \frac{\mathsf{Number\ of\ target\ outcomes}}{\mathsf{Number\ of\ all\ outcomes}}\]

\[P = \frac{72}{200} = .36\]

Risks/Probabilities (cont.)

• Creating an example in R using RStudio

pain.df <- data.frame(Ethnicity = c("Latin", "Non-Latin"),
                      Diagnosed = c(72, 223),
                      Not_Diagnosed = c(128, 241)
                     )

Risks/Probabilities (cont.)

library(knitr)
library(kableExtra)
table.of.pain <- pain.df %>%
  kable(col.names = c("Ethnicity", "Diagnosed", "Not Diagnosed"),
        align = c("l", "c", "c")) %>%
  kable_styling(font_size = 42) %>% kable_material("hover") %>% 
  add_header_above(c(" " = 1, "Pain Disorder Diagnosis" = 2))

Risks/Probabilities (cont.)

• Simply giving the name of an object invokes it
  • (Kinda like Satan)

table.of.pain

Risks/Probabilities (end)

\[P(\mathsf{Latino\ Patient\ Has\ a\ Pain\ Disorder}) = \frac{\mathsf{Target\ Event}}{\mathsf{All\ Events}}\]

\[= \frac{\mathsf{Latinos\ \overline{c}\ Pain\ Disorders}}{(\mathsf{Latinos\ \overline{c}\ Pain\ Disorders})+(\mathsf{Latinos\ \overline{s}\ Pain\ Disorders})}\]

\[= \frac{72}{72+128} = \frac{72}{200} = .36\]

More Computations with R

• Brackets ([]) index parts (“elements”) of an object

pain.df[1, 1] # First row, first column

## [1] "Latin"

pain.df[1, 2] # First row, second column

## [1] 72

pain.df[1, ] # The whole first row

##   Ethnicity Diagnosed Not_Diagnosed
## 1     Latin        72           128

More Computations with R (cont.)

pain.df

##   Ethnicity Diagnosed Not_Diagnosed
## 1     Latin        72           128
## 2 Non-Latin       223           241

pain.df[1, 2] + pain.df[1, 3]

## [1] 200

p.latin.patient.diagnosed.with.pain.disorder <- pain.df[1, 2] / (pain.df[1, 2] + pain.df[1, 3])
p.latin.patient.diagnosed.with.pain.disorder

## [1] 0.36

More Computations with R (cont.)

• $ accesses a column (or other variable-like element) of an object…

pain.df$Ethnicity

## [1] "Latin"     "Non-Latin"

pain.df$Diagnosed

## [1]  72 223

More Computations with R (cont.)

• We can do computations etc. on these columns

pain.df$Diagnosed / (pain.df$Diagnosed + pain.df$Not_Diagnosed)

## [1] 0.3600000 0.4806034

More Computations with R (cont.)

• And assign those computations to an element
  • That’s also part of that data frame

pain.df$Risk_of_Pain_Diagnosis <- pain.df$Diagnosed /
          (pain.df$Diagnosed + pain.df$Not_Diagnosed)
pain.df$Risk_of_Pain_Diagnosis

## [1] 0.3600000 0.4806034

pain.df

##   Ethnicity Diagnosed Not_Diagnosed Risk_of_Pain_Diagnosis
## 1     Latin        72           128              0.3600000
## 2 Non-Latin       223           241              0.4806034

More Computations with R (end)

• The risk ratio is simply the ratio of these risks:

\[\mathsf{Risk\ Ratio} = \frac{\mathsf{Risk\ Among\ Latino\ Patients}}{\mathsf{Risk\ Among\ Non\text{-}Latino\ Patients}}\]

\[= \frac{.36}{\sim.48} \approx .75 \]

risk_ratio <- pain.df$Risk_of_Pain_Diagnosis[1] /
              pain.df$Risk_of_Pain_Diagnosis[2]
risk_ratio

## [1] 0.7490583

Hazards & Hazard Ratios

• Hazard is simply risk within some time frame
  • “What are the risks of relapse within one year?”
• Hazard ratio is the hazard in one group relative to an other
  • “What are the risks of Latino patients relapsing within one year compared to non-Latino patients?”

Presentation of Hazards

• Since events in a time frame may not be uniform,
  • Hazards / hazard ratios are often accompanied by “Kaplan-Meier curves” depicting events over time

Analyzing Hazards

• Cox proportional hazards regression
  • Tests if variables can significantly predict hazard (time-to-event)
  • Handles “censoring” well
    • When we don’t know how long it was to an event because, e.g., the event didn’t happen during the study
      • (E.g., the patient survived until at least the end of the study window)
  • Assumes that the effect of each predictor on the outcome is consistent throughout the study
    • E.g., there is no seasonal effect

Example of Cox Regression

Example of Cox Regression (cont.)

Example of Cox Regression (cont.)

Example of Cox Regression (end)

Risks vs. Odds

• Probability is the chance of something happening out of all possible occasions
  • E.g., of the 200 Latino patients with OUDs, 72 were diagnosed with pain disorders

\[P(\mathsf{Being\ Diagnosed\ \overline{c}\ Pain\ Disorder})= \frac{72}{72+128} = \frac{72}{200}\]

• I.e., \(\frac{72}{200} = .36 \approx \frac{1}{3}\) of all Latino patients with OUDs were diagnosed with a pain disorder

Risks vs. Odds (cont.)

• Odds are the chance of something happening relative to them not happening
  • E.g., 72 Latino patients were diagnosed with pain disorders
  • 128 Latino patients were not so diagnosed

\[\mathsf{Odds\ of\ Being\ Diagnosed\ \overline{c}\ Pain\ Disorder} = \frac{72}{128} \approx .56\]

Examples of Odds

Odds Ratios

• Odds ratios (OR) represent the odds of an event in one group proportional to the odds in an other
  • E.g., OR = .5 means the odds in the target group are half the odds in the reference group
    • E.g., the odds in the target group are 2:1
      • While the odds in the reference group are 4:1
• Turning back to the OUD study may help…

Odds Ratios: Example

Odds—not odds ratios—for each group:

• Odds of pain diagnosis among non-Latino patients with OUDs: \(\frac{N_{\mathsf{non-Latinos\ \overline{c}\ Pain\ Disorders}}}{N_{\mathsf{non-Latinos\ \overline{s}\ Pain\ Disorders}}} = \frac{223}{241} \approx .92\)

Odds Ratios: Equation

\[OR = \frac{(\textsf{Target & Present / Target & Not Present})}{(\textsf{Reference & Present / Reference & Not Present})}\]

\[OR = \frac{(\textsf{A / B)}}{\textsf{(C / D)}}\]

Odds Ratios: Example (cont.)

\[OR = \frac{(\textsf{Latin & Diagnosed / Latin & Not Diagnosed})}{(\textsf{Non-Latin & Diagnosed / Non-Latin & Not Diagnosed})}\]

\[OR = \frac{(72 / 128)}{(223 / 241)} \approx \frac{.56}{.92}\approx .61\]

Odds Ratios: Example (cont.)

• Could also look at it from the non-Latin perspective

\[OR = \frac{(223 / 241)}{(72 / 128)} = \frac{.92}{.56} \approx 1.6\]

Note that \(\frac{1}{.61} \approx 1.6\)

• Non-Latino OUD patients are 1.6 times more likely than Latino patients to be diagnosed with a pain disorder

Contingency Tables

Contingency Tables (cont.)

Yeah, like this:

• Odds among Latinos = \(\frac{72}{128} \approx .56\)
• Odds among non-Latinos = \(\frac{223}{241} \approx .92\)
• Odds ratio = \(\frac{.56}{.92} \approx .61\)

Fisher’s Exact Test

• Invented to test if Fisher’s colleague, Muriel Bristol, could indeed tell if cream or tea were poured first into a cup
• Called an “exact” test because it computes the exact p-value for rejection
  • I.e., not an estimate based on
    inferences about the population
    (e.g., normality)
  • So, isn’t inferential per se

Fisher’s Exact Test Example

pain.df.counts <- subset(pain.df, select = c("Diagnosed", "Not_Diagnosed"))
fisher.test(pain.df.counts)

## 
##  Fisher's Exact Test for Count Data
## 
## data:  pain.df.counts
## p-value = 0.00491
## alternative hypothesis: true odds ratio is not equal to 1
## 95 percent confidence interval:
##  0.4250886 0.8665430
## sample estimates:
## odds ratio 
##  0.6083658

Fisher’s Exact Test Example (cont.)

• fisher.test also creates an object
• And not all elements of that object are automatically printed
• We can look to see what all is there with either the powerful & flexible summary() command:

summary(fisher.test(pain.df.counts))

##             Length Class  Mode     
## p.value     1      -none- numeric  
## conf.int    2      -none- numeric  
## estimate    1      -none- numeric  
## null.value  1      -none- numeric  
## alternative 1      -none- character
## method      1      -none- character
## data.name   1      -none- character

Fisher’s Exact Test Example (cont.)

• Or with the more revealing str() command:
  • str() works to “look inside” most any object or function in R

str(fisher.test(pain.df.counts))

## List of 7
##  $ p.value    : num 0.00491
##  $ conf.int   : num [1:2] 0.425 0.867
##   ..- attr(*, "conf.level")= num 0.95
##  $ estimate   : Named num 0.608
##   ..- attr(*, "names")= chr "odds ratio"
##  $ null.value : Named num 1
##   ..- attr(*, "names")= chr "odds ratio"
##  $ alternative: chr "two.sided"
##  $ method     : chr "Fisher's Exact Test for Count Data"
##  $ data.name  : chr "pain.df.counts"
##  - attr(*, "class")= chr "htest"

Fisher’s Exact Test Example (cont.)

• $ can access any of those elements of the fisher.test object:

fisher.test(pain.df.counts)$p.value

## [1] 0.004910079

fisher.test(pain.df.counts)$conf.int

## [1] 0.4250886 0.8665430
## attr(,"conf.level")
## [1] 0.95

Is Fisher’s Exact Test Too Conservative?

• Fisher himself sure was
• Fisher’s exact test can mis-estimate p-values when the total N is “small” (<50; Andrés & Tejedor, 1995)
  • But this bias itself is usually small
  • And common corrections (e.g., Yate’s correction) don’t change outcomes much (Crans & Shuster, 2008)
    • While making for somewhat more complicated analyses that rely on parametric assumptions

Upton (1992)

Upton (1992, cont.)

Normal Distribution

Formula for a normal distribution:

\[f(x, \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x - \mu)^2}{2\sigma^2}}\]

• x is some random variable
• μ is the mean
• σ is the standard deviation

• There, now you can say you learned this

Characteristics

Characteristics (cont.)

• Most importantly, it is only a function of the mean & standard deviation
• The mean, median and mode are all equal
• The total area under the curve equals 1
• It’s symmetric
• The curve approaches—but never touches—the x-axis

Q-Q Plots

More Figures, More Views on Skew

From StackExchange

Q-Q Plots (cont.)

• Can put a “confidence envelope” around data in Q-Q plots:

Alternatives to Q-Q Plots

Background

• Invented by Karl Pearson (in an abstruse 1900 article)
• Originally to test “goodness of fit”
  • I.e., how well a set of data fit a theoretical distribution
  • Or if two sets of data follow the same distribution
    
    
• Technically, χ² is a type of Gamma distribution created by the distribution of sums of the squares of a set of standard normal random variables
  • A lot like we get when computing ordinary least squares for t-tests & ANOVAs
  • And is closely related to the t and F distributions

Characteristics

Characteristics (cont.)

Computing χ²

Example of Using χ²

• Remember our data for Latino patients with OUDs:

Example of Using χ² (cont.)

Example of Using χ² (end)

qchisq(df = 1, p = .05, lower.tail = FALSE)

## [1] 3.841459

• Which is smaller than our χ² of 15.68
• So our observed values are significantly different than our expected ones

Uses of χ²

• Because it only depends on df,
  • And resembles a normal distribution.
• It is useful for testing if data follow a normal distribution
  • Or often if the total set of deviations from normality
    • (Or any set of expected values)
  • Are greater than expected
    
    
• It can do this for discrete values—like counts
  • t and F distributions technically can’t do this

Uses of χ² (cont.)

• The χ² distribution has many uses, including:
  1. Estimating of parameters of a population of an unknown distribution
  2. Checking the relationships between categorical variables
  3. Checking independence of two criteria of classification of multiple qualitative variables
  4. Testing deviations of differences between expected and observed frequencies
  5. Conducting goodness of fit tests

Example of a χ² Test