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  • 2. Statistics
    • Built-in support for statistics
    • Distribution functions
    • Example
    • Create a normal distribution curve
    • Simple testing
    • Example analysis
    • Linear Regression
    • Exercise: Exercise 6
    • More words of warning

2. Statistics

Built-in support for statistics

  • R is a statistical programming language:
    • Classical statistical tests are built-in
    • Statistical modeling functions are built-in
    • Regression analysis is fully supported
    • Additional mathematical packages are available (MASS, Waves, sparse matrices, etc)

Distribution functions

  • Most commonly used distributions are built-in, functions have stereotypical names, e.g. for normal distribution:
    • pnorm - cumulative distribution for x
    • qnorm - inverse of pnorm (from probability gives x)
    • dnorm - distribution density
    • rnorm - random number from normal distribution
distributions

distributions

  • Available for variety of distributions: punif (uniform), pbinom (binomial), pnbinom (negative binomial), ppois (poisson), pgeom (geometric), phyper (hyper-geometric), pt (T distribution), pf (F distribution)

  • 10 random values from the Normal distribution with mean 10 and standard deviation 5:

rnorm(10, mean=10, sd=5)
 [1]  5.693620  4.429255  0.442925  9.223783 14.217208 10.117813 10.052955  9.347432
 [9]  7.879201  3.260119
  • The probability of drawing exactly 10 from this distribution:
dnorm(10, mean=10, sd=5)
[1] 0.07978846
dnorm(100, mean=10, sd=5)
[1] 3.517499e-72
  • The probability of drawing a value smaller than 10:
pnorm(10, mean=10, sd=5)
[1] 0.5
  • The inverse of pnorm():
qnorm(0.5, mean=10, sd=5)
[1] 10
  • How many standard deviations for statistical significance?
qnorm(0.95, mean=0, sd=1)
[1] 1.644854

Example

Recall our histogram of Wind Speed from yesterday:

  • The data look to be roughly normally-distributed
  • An assumption we rely on for various statistical tests
weather <- read.csv("ozone.csv")
hist(weather$Wind, col="steelblue", xlab="Wind Speed",
     main="Distribution of Wind Speed",
     breaks = 20, freq=FALSE)

Create a normal distribution curve

  • If our data are normally-distributed, we can calculate the probability of drawing particular values.
    • e.g. a Wind Speed of 10
windMean <- mean(weather$Wind)
windSD <- sd(weather$Wind)
dnorm(10, mean=windMean, sd=windSD)
[1] 0.1132311
  • We can overlay this on the histogram using points as we just saw:
hist(weather$Wind, col="steelblue", xlab="Wind Speed",
     main="Distribution of Wind Speed",
     breaks = 20, freq=FALSE)
points(10, dnorm(10, mean=windMean, sd=windSD),
       col="red", pch=16)

  • We can repeat the calculation for a vector of values
    • remember that functions in R are often vectorized
    • use lines in this case rather than points
xs <- c(0,5,10,15,20)
ys <- dnorm(xs, mean=windMean, sd=windSD)
hist(weather$Wind, col="steelblue", xlab="Wind Speed",
     main="Distribution of Wind Speed",
     breaks = 20, freq=FALSE)
lines(xs, ys, col="red")

  • For a smoother curve, use a longer vector:
    • we can generate x values using the seq() function
  • Inspecting the data in this manner is usually acceptable to assess normality
    • the fit doesn’t have to be exact
    • the shapiro test is also available ?shapiro.test (but not really recommended by statisticians)
hist(weather$Wind,col="steelblue",xlab="Wind Speed",
     main="Distribution of Wind Speed",breaks = 20,freq=FALSE)
xs <- seq(0,20, length.out = 10000)
ys <- dnorm(xs, mean=windMean,sd=windSD)
lines(xs,ys,col="red")

Simple testing

  • If we want to compute the probability of observing a particular Wind Speed, from the same distribution, we can use the standard formula to calculate a t statistic:

t=ˉx−μ0s/√(n)

  • Say a Wind Speed of 2; which from the histogram seems to be unlikely
t <- (windMean - 2) / (windSD/sqrt(length(weather$Wind)))
t
[1] 27.93897
  • …or use the t.test() function to compute the statistic and corresponding p-value
t.test(weather$Wind, mu=2)

    One Sample t-test

data:  weather$Wind
t = 27.939, df = 152, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 2
95 percent confidence interval:
  9.394804 10.520229
sample estimates:
mean of x 
 9.957516
  • A variety of tests are supported the R authors have tried to make them as consistent as possible
?var.test
?t.test
?wilcox.test
?prop.test
?cor.test
?chisq.test
?fisher.test

Example analysis

  • We have already seen that men in our patients dataset tend to be heavier than women
  • We can test this formally in R
    • N.B. the weight of people in a population tends to be normally distributed, so we can probably be safe to use a parametric test

We need to run this if we don’t have the patients data in our R environment

patients <- read.delim("patient-info.txt")

We can test if the variance of the two groups is the same

  • this will influence what flavour of t-test we use
var.test(patients$Weight~patients$Sex)

    F test to compare two variances

data:  patients$Weight by patients$Sex
F = 0.14216, num df = 49, denom df = 44, p-value = 3.59e-10
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
 0.07900337 0.25344664
sample estimates:
ratio of variances 
         0.1421572 
t.test(patients$Weight~patients$Sex, var.equal=FALSE)

    Welch Two Sample t-test

data:  patients$Weight by patients$Sex
t = -11.204, df = 55.168, p-value = 7.786e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -15.62079 -10.88094
sample estimates:
mean in group Female   mean in group Male 
            68.95980             82.21067

If were unwilling to make a assumption of normality, a non-parametric test could be used.

wilcox.test(patients$Weight~patients$Sex)

    Wilcoxon rank sum test with continuity correction

data:  patients$Weight by patients$Sex
W = 59, p-value = 1.993e-15
alternative hypothesis: true location shift is not equal to 0

Linear Regression

  • Linear modeling is supported by the function lm():
    • example(lm)
  • The output assumes you know a fair bit about the subject
  • lm is really useful for plotting lines of best fit to XY data, in order to determine intercept, gradient and Pearson’s correlation coefficient
    • This is very easy in R

  • Three steps to plotting with a best fit line:

    • Plot XY scatter-plot data
    • Fit a linear model
    • Add bestfit line data to plot with abline() function

  • Let’s see a toy example:-

x <- c(1, 2.3, 3.1, 4.8, 5.6, 6.3)
y <- c(2.6, 2.8, 3.1, 4.7, 5.1, 5.3)
plot(x,y, xlim=c(0,10), ylim=c(0,10))

  • The ~ is used to define a formula; i.e. “y is given by x”
    • Take care about the order of x and y in the plot and lm expressions
plot(x,y, xlim=c(0,10), ylim=c(0,10))
myModel <- lm(y~x)
abline(myModel, col="red")

The generic summary function give an overview of the model

  • particular components are accessible if we know their names
summary(myModel)

Call:
lm(formula = y ~ x)

Residuals:
       1        2        3        4        5        6 
 0.33159 -0.22785 -0.39520  0.21169  0.14434 -0.06458 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  1.68422    0.29056   5.796   0.0044 **
x            0.58418    0.06786   8.608   0.0010 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3114 on 4 degrees of freedom
Multiple R-squared:  0.9488,    Adjusted R-squared:  0.936 
F-statistic:  74.1 on 1 and 4 DF,  p-value: 0.001001
names(myModel)  # Names of the objects within myModel
 [1] "coefficients"  "residuals"     "effects"       "rank"          "fitted.values"
 [6] "assign"        "qr"            "df.residual"   "xlevels"       "call"         
[11] "terms"         "model"
  • alternatively, various helper functions are provided.
coef(myModel)   # Coefficients
(Intercept)           x 
  1.6842239   0.5841843 
resid(myModel)  # Residuals
          1           2           3           4           5           6 
 0.33159186 -0.22784770 -0.39519512  0.21169160  0.14434418 -0.06458482 
fitted(myModel) # Fitted values
       1        2        3        4        5        6 
2.268408 3.027848 3.495195 4.488308 4.955656 5.364585 
residuals(myModel) + fitted(myModel) # what values does this give?
  1   2   3   4   5   6 
2.6 2.8 3.1 4.7 5.1 5.3

You can also get some diagnostic information on the model.

  • Some explanation can be found here and here
par(mfrow=c(2,2)) 
plot(myModel)

  • R has a very powerful formula syntax for describing statistical models
  • Suppose we had two explanatory variables x and z, and one response variable y
  • We can describe a relationship between, say, y and x using a tilde ~, placing the response variable on the left of the tilde and the explanatory variables on the right:
    • y~x
  • It is very easy to extend this syntax to do multiple regressions, ANOVAs, to include interactions, and to do many other common modelling tasks. For example
y~x       #If x is continuous, this is linear regression
y ~ x
y~x       #If x is categorical, ANOVA
y ~ x
y~x+z     #If x and z are continuous, multiple regression
y ~ x + z
y~x+z     #If x and z are categorical, two-way ANOVA
y ~ x + z
y~x+z+x:z # : is the symbol for the interaction term
y ~ x + z + x:z
y~x*z     # * is a shorthand for x+z+x:z
y ~ x * z

Exercise: Exercise 6

  • There are suggestions that Ozone level could be influenced by Temperature:
plot(weather$Temp, weather$Ozone,xlab="Temperature",ylab="Ozone level",pch=16)

  • Perform a linear regression analysis to assess this:
    • Fit the linear model and print a summary of the output
    • Plot the two variables and overlay a best-fit line
    • What is the equation of the best-fit line in the form y=ax+b
  • Calculate the correlation between the two variables using the function cor (?cor)
    • can you add text to the plot to show the correlation coefficient?
    • HINT: The paste can be used to join strings of text together, or variables
paste("Hello","World")
[1] "Hello World"
age <- 35
paste("My age is", age)
[1] "My age is 35"
## Your Answer Here ##

More words of warning

Correlation != Causation

http://tylervigen.com/spurious-correlations

So if I want to win a nobel prize, I should eat even more chocolate?!?!?

But no-one would ever take such trends seriously….would they?

Cutting-down on Ice Cream was recommended as a safeguard against polio!

Cutting-down on Ice Cream was recommended as a safeguard against polio!

---
title: "Introduction to Solving Biological Problems Using R - Day 2"
author: Mark Dunning, Suraj Menon and Aiora Zabala. Original material by Robert Stojnić,
  Laurent Gatto, Rob Foy, John Davey, Dávid Molnár and Ian Roberts
date: '`r format(Sys.time(), "Last modified: %d %b %Y")`'
output:
  html_notebook:
    toc: yes
    toc_float: yes
---


# 2. Statistics
##Built-in support for statistics
- R is a statistical programming language:
    + Classical statistical tests are built-in
    + Statistical modeling functions are built-in
    + Regression analysis is fully supported
    + Additional mathematical packages are available (`MASS`, Waves, sparse matrices, etc)
  
##Distribution functions  
- Most commonly used distributions are built-in, functions have stereotypical names, e.g. for normal distribution:
    + **`pnorm`** - cumulative distribution for x
    + **`qnorm`** - inverse of pnorm (from probability gives x)
    + **`dnorm`** - distribution density
    + **`rnorm`** - random number from normal distribution
  
![distributions](images/distributions.png)  
  
- Available for variety of distributions: `punif` (uniform), `pbinom` (binomial), `pnbinom` (negative binomial), `ppois` (poisson), `pgeom` (geometric), `phyper` (hyper-geometric), `pt` (T distribution), pf (F distribution) 


- 10 random values from the Normal distribution with mean 10 and standard deviation 5:
```{r}
rnorm(10, mean=10, sd=5)
```
- The probability of drawing exactly 10 from this distribution:
```{r}
dnorm(10, mean=10, sd=5)
```

```{r}
dnorm(100, mean=10, sd=5)
```


- The probability of drawing a value smaller than 10:
```{r}
pnorm(10, mean=10, sd=5)
```
- The inverse of `pnorm()`:
```{r}
qnorm(0.5, mean=10, sd=5)
```
- How many standard deviations for statistical significance?
```{r}
qnorm(0.95, mean=0, sd=1)
```

## Example

Recall our histogram of Wind Speed from yesterday:

- The data look to be roughly normally-distributed
- An assumption we rely on for various statistical tests

```{r}
weather <- read.csv("ozone.csv")
hist(weather$Wind, col="steelblue", xlab="Wind Speed",
     main="Distribution of Wind Speed",
     breaks = 20, freq=FALSE)
```

## Create a normal distribution curve

- If our data are normally-distributed, we can calculate the probability of drawing particular values.
      + e.g. a Wind Speed of 10

```{r}
windMean <- mean(weather$Wind)
windSD <- sd(weather$Wind)
dnorm(10, mean=windMean, sd=windSD)
```

- We can overlay this on the histogram using `points` as we just saw:
```{r}
hist(weather$Wind, col="steelblue", xlab="Wind Speed",
     main="Distribution of Wind Speed",
     breaks = 20, freq=FALSE)
points(10, dnorm(10, mean=windMean, sd=windSD),
       col="red", pch=16)
```

- We can repeat the calculation for a vector of values
    + remember that functions in R are often ***vectorized***
    + use `lines` in this case rather than `points`
    
```{r}
xs <- c(0,5,10,15,20)
ys <- dnorm(xs, mean=windMean, sd=windSD)
hist(weather$Wind, col="steelblue", xlab="Wind Speed",
     main="Distribution of Wind Speed",
     breaks = 20, freq=FALSE)
lines(xs, ys, col="red")
```




- For a smoother curve, use a longer vector:
    + we can generate x values using the `seq()` function
- Inspecting the data in this manner is usually acceptable to assess normality
    + the fit doesn't have to be exact
    + the shapiro test is also available `?shapiro.test` (but not really recommended by statisticians)


```{r}
hist(weather$Wind,col="steelblue",xlab="Wind Speed",
     main="Distribution of Wind Speed",breaks = 20,freq=FALSE)
xs <- seq(0,20, length.out = 10000)
ys <- dnorm(xs, mean=windMean,sd=windSD)
lines(xs,ys,col="red")
```

## Simple testing

- If we want to compute the probability of observing a particular Wind Speed, from the same distribution, we can use the standard formula to calculate a t statistic:

$$t = \frac{\bar{x} -\mu_0}{s / \sqrt(n)}$$

- Say a Wind Speed of 2; which from the histogram seems to be unlikely

```{r}
t <- (windMean - 2) / (windSD/sqrt(length(weather$Wind)))
t
```

- ...or use the **`t.test()`** function to compute the statistic and corresponding p-value

```{r}
t.test(weather$Wind, mu=2)
```

- A variety of tests are supported the R authors have tried to make them as consistent as possible

```{r}
?var.test
?t.test
?wilcox.test
?prop.test
?cor.test
?chisq.test
?fisher.test
```

    
- Bottom-line: Pretty much any statistical test you care to name will probably be in R
    + This is not supposed to be a statistics course (sorry!)
    + None of them are particular harder than others to use
    + The difficulty is deciding which test to use:
        + whether the assumptions of the test are met, etc.
    + Consult your local statistician if not sure
    + An upcoming course that will help
        + [Introduction to Statistical Analysis](http://bioinformatics-core-shared-training.github.io/IntroductionToStats/)
    + Some good references:
        + [Statistical Analysis Using R (Course from the Babaraham Bioinformatics Core)](http://training.csx.cam.ac.uk/bioinformatics/event/1827771)
        + [Quick-R guide to stats](http://www.statmethods.net/stats/index.html)
        + [Simple R eBook](https://cran.r-project.org/doc/contrib/Verzani-SimpleR.pdf)
        + [R wiki](https://en.wikibooks.org/wiki/R_Programming/Descriptive_Statistics)
        + [R tutor](http://www.r-tutor.com/elementary-statistics)
        + [Statistical Cheatsheet](https://rawgit.com/bioinformatics-core-shared-training/intermediate-stats/master/cheatsheet.pdf)
   - R will do whatever you ask it, it will never check the assumptions or help you to interpret the result

![](images/clippy.jpg)

## Example analysis

- We have already seen that men in our `patients` dataset tend to be heavier than women
- We can **test this formally** in R
    + N.B. the weight of people in a population tends to be normally distributed, so we can probably be safe to use a parametric test
    
![](images/males-versus-females.png)




We need to run this if we don't have the patients data in our R environment
```{r}
patients <- read.delim("patient-info.txt")
```

We can test if the variance of the two groups is the same

+ this will influence what flavour of t-test we use

```{r}
var.test(patients$Weight~patients$Sex)
```


```{r}
t.test(patients$Weight~patients$Sex, var.equal=FALSE)
```

If were unwilling to make a assumption of normality, a non-parametric test could be used.

```{r}
wilcox.test(patients$Weight~patients$Sex)
```

## Linear Regression

- Linear modeling is supported by the function `lm()`:
    + `example(lm)`
- The output assumes you know a fair bit about the subject
- `lm` is really useful for plotting lines of best fit to XY data, in order to determine intercept, gradient and Pearson's correlation coefficient
    + This is very easy in R
- Three steps to plotting with a best fit line:

    + Plot XY scatter-plot data
    + Fit a linear model
    + Add bestfit line data to plot with `abline()` function
- Let's see a toy example:-

```{r}
x <- c(1, 2.3, 3.1, 4.8, 5.6, 6.3)
y <- c(2.6, 2.8, 3.1, 4.7, 5.1, 5.3)
plot(x,y, xlim=c(0,10), ylim=c(0,10))
```

- The `~` is used to define a formula; i.e. "y is given by x"
    + Take care about the order of `x` and `y` in the `plot` and `lm` expressions

```{r}
plot(x,y, xlim=c(0,10), ylim=c(0,10))
myModel <- lm(y~x)
abline(myModel, col="red")
```

The generic `summary` function give an overview of the model

- particular components are accessible if we know their names

```{r}
summary(myModel)
names(myModel)  # Names of the objects within myModel
```

- alternatively, various helper functions are provided.

```{r}
coef(myModel)   # Coefficients
resid(myModel)  # Residuals
fitted(myModel) # Fitted values
residuals(myModel) + fitted(myModel) # what values does this give?
```

You can also get some diagnostic information on the model.

- Some explanation can be found [here](http://data.library.virginia.edu/diagnostic-plots/) and [here](http://strata.uga.edu/6370/rtips/regressionPlots.html)

```{r}
par(mfrow=c(2,2)) 
plot(myModel)
```

- R has a very powerful formula syntax for describing statistical models
- Suppose we had two explanatory variables `x` and `z`, and one response variable `y`
- We can describe a relationship between, say, `y` and `x` using a tilde `~`, placing the response variable on the left of the tilde and the explanatory variables on the right:
    + y~x
- It is very easy to extend this syntax to do multiple regressions, ANOVAs, to include interactions, and to do many other common modelling tasks. For example

```{r}
y~x       #If x is continuous, this is linear regression
y~x       #If x is categorical, ANOVA
y~x+z     #If x and z are continuous, multiple regression
y~x+z     #If x and z are categorical, two-way ANOVA
y~x+z+x:z # : is the symbol for the interaction term
y~x*z     # * is a shorthand for x+z+x:z
```

## Exercise: Exercise 6

- There are suggestions that Ozone level could be influenced by Temperature:
```{r}
plot(weather$Temp, weather$Ozone,xlab="Temperature",ylab="Ozone level",pch=16)
```

- Perform a linear regression analysis to assess this:
    + Fit the linear model and print a summary of the output
    + Plot the two variables and overlay a best-fit line
    + What is the equation of the best-fit line in the form
      $$ y = ax + b$$
- Calculate the correlation between the two variables using the function cor (?cor)
    + can you add text to the plot to show the correlation coefficient?
    + HINT: The `paste` can be used to join strings of text together, or variables

```{r}
paste("Hello","World")
age <- 35
paste("My age is", age)
```


```{r}

## Your Answer Here ##

```

![](images/exercise6.png)
![](images/exercise6b.png)


## More words of warning

***Correlation != Causation***

![](images/spurious.png)

http://tylervigen.com/spurious-correlations


![](images/nobel-prize.jpeg)

[So if I want to win a nobel prize, I should eat even more chocolate?!?!?](http://www.businessinsider.com/chocolate-consumption-vs-nobel-prizes-2014-4?IR=T)

But no-one would ever take such trends seriously....would they?


![Cutting-down on Ice Cream was recommended as a safeguard against polio!](images/icecreamvspolio.jpg)

