Download Quantitative analysis and R – (1)

Quantitative analysis and R – (1)
LING115
November 18, 2009
Some basic statistics
Reference
The basics
• Measures of central tendency, dispersion
• Frequency distribution
• Hypothesis testing
– Population vs. sample
– One sample t-test
• Measures of association
– Covariance and correlation
Observations
• Our linguistic data will consist of a set of
observations
• Each observation describes some property of
a linguistic entity of our research interest
– F1 of the English vowel /i/
– The word that appears before ‘record’ used as a
verb
– Grammaticality of ‘Colorless green ideas sleep
furiously’
Measures of central tendency
• Median
– The value in the middle assuming that the values
in the data are ordered according to their size
• Mode
– The most frequent value in the data
• Mean
– The arithmetic mean of values in the data
Measures of dispersion
• Deviation
– Difference between a value and a measure of
central tendency (e.g. mean)
• Variance
– Average of sum of squared deviation from the
mean
• Standard deviation
– The square root of variance
Frequency distribution
• Distribution describing how often each value
of an observation occurs in the data
• Enumerating the frequency of each value of
an observation may not be informative,
especially if the observations can have
continuous values
• Instead we can characterize the frequency
distribution in terms of ranges of value
Histogram
• Define bins, or
contiguous ranges of
values
• Put the observations
into bins
• Plot the number of
observations that
belong to each bin
Histograms with smaller bins
Continuous curve and probability
• As the bin gets smaller, the histogram looks more
like a continuous curve
• Once we interpret a histogram as a continuous
curve, it makes more sense to calculating the
probability that the observations falls within a
range of values rather than counting the number
of such observations
• The probability is the ratio of the area under the
curve within the given range to the total area
under the curve
Uniform distribution
Bimodal distribution
Normal distribution
Skewed distribution
Normal distribution
• Symmetric bell-shaped curve
– Mean = median = mode
• The distribution can be solely defined in terms of
the mean and the standard deviation
– Mean (μ) defines the center of the curve
– Standard deviation (σ)defines the spread of the curve
– N(μ, σ) means a normal distribution whose mean= μ,
standard deviation=σ
– N(0,1) is called the standard normal distribution
Z-score
• Z-score measures the distance of a value from
the mean in terms of standard deviation units
– Subtract the mean from the value
– Normalize the distance by the standard deviation
– i.e. z  xi  

• Calculating the z-score for every value of a
normal distribution converts the distribution
into a standard normal distribution
Standard normal (Z) table
• Recall that we calculate the probability of a value
falling within a particular range by calculating the
area under the curve
• To skip the calculation part, people have provided
distribution tables for some popular distributions
• The standard normal distribution is one of them
• http://www.statsoft.com/textbook/sttable.html
Population vs. sample
• Population
– The entire set
– e.g. The set of all people who live in California
– e.g. The set of all sentences in English
• Sample
– A subset of the population
– e.g. A set of 50,000 people who live in California
– e.g. The set of sentences in the WSJ corpus
Sample
• We analyze a sample when we examine a
corpus
• We hope our sample is a good representation
of the population
• Otherwise we cannot generalize a statistical
tendency found in a corpus to make claims
about the language
A good sample
• Size
– The sample must be large enough
• Randomness
– Members of the sample must be chosen randomly
from the population
Sample statistics
• Statistics about a sample is an estimation of
the population parameter with possible errors
due to sampling
Degree of freedom
• Degrees of freedom reflect how precise our
estimation is
– The bigger the size of a sample, the more precise
our estimation of the population parameter
• Initially, degrees of freedom is equal to the
size of the sample
• Degrees of freedom decrease as we estimate
more parameters with the same data
Measures – revisited
• Mean
– Sample mean:
n
x
– Population mean:
• Variance
– Sample variance:
x
i 1
i
n
N

x
i 1
i
N
n
s2 
– Population variance:
 (x  x)
i 1
2
i
(n  1)
N
2 
 (x  x)
i 1
2
i
N
Central limit theorem
• As the number of observations in each sample
increases, the means of the samples tend
toward the normal distribution
• http://www.stat.sc.edu/~west/javahtml/CLT.html
• The applet actually illustrates that the sum of
dice converges to normality, but this also
applies to the sample means since we can
divide the sum by the number of dice
Standard error (SE)
• Standard deviation of means of samples of a population
• Intuitively, this would be calculated by first sampling the
data from the population many times and then
calculating the standard deviation of the means
• There is a way to directly calculate the standard error
from the standard deviation of the population or the
sample

– From population:
SE 
– From sample:
sx
SE 
n
n
Comparing means – (1)
• Question
– We do expect the sample mean to be somewhat
different even if the samples are from the same
population
– But then how do we tell if the mean of a data set
is too different to say that the data set is from a
different population?
Comparing means – (2)
• Basic idea
– The goal is to define what we mean by “the mean
is too different”
– The distribution of sample means of a population
follows the normal distribution
– We measure the distance of a sample mean from
the population mean in terms of standard error
– The farther away from the population mean, the
less likely it is that the sample is from the given
population
One sample t-test – (1)
• t-score measures deviation of a sample mean
from a given population mean in terms of
standard error
x
t
sx
• This is just like converting a sample value to
the z-score, except that the sample value here
is the sample mean
One sample t-test – (2)
• The distribution of tscores looks like the
standard normal
distribution, N(0,1)
• The larger the size of a
sample, the closer the tdistribution is to the
standard normal
distribution
One sample t-test – (3)
• Once we have the t-score (t), we ask “how
likely is it to get a value less/greater than or
equal to t from the t-distribution?”
• We can answer this by calculating the relevant
area under the curve or looking up the t-table
• If you think the probability is too small, you
have reason to suspect that your sample
mean is not from the distribution of possible
sample means of a population
A more typical way to put it
• Null hypothesis (H0): your sample mean is not different
from the population mean (the apparent difference is
simply due to error inherent in the sampling process)
• We decide whether to accept or reject the null
hypothesis by performing one-sample t-test
• Let’s say α is the probability that the t-score
representing the sample mean is from the tdistribution representing the distribution of sample
means of the population
• If α is smaller than some threshold we predefined (e.g.
0.5) , we reject the null hypothesis
A more typical way to put it – (2)
• Note that unless α is zero, we can never be
confident rejecting the null hypothesis is the right
thing
• We call the error of falsely rejecting the null
hypothesis “Type-I error”
• α is the probability that we will commit type-I
error
• 1- α is the probability that we won’t
• We can say “we are (1- α)*100 percent confident
that the null-hypothesis is wrong”
Measures of association
• Question
– We want to see if two variables are related to
each other
• Basic idea
– If the values of two variables fluctuate in the same
direction, in similar magnitudes, they are probably
related
– Degree of fluctuation is measured in terms of the
deviation from the mean
Covariance
• Average sum of product of deviations
n
 ( x  x )  ( y  y)
i 1
i
i
n
• If x and y fluctuate in the same direction, in similar
magnitudes, the sum of product of deviations will be large
• The sum of product will be larger if we have more pairs to
compare
• This is not desirable, so we normalize the sum by the number
of pairs
Correlation
• Same as covariance except that the deviation
is measured in terms of z-scores
( xi  x ) ( yi  y )


sx
sy
i 1
n
n
• The idea is to make the magnitudes of
deviation comparable by putting both x and y
on the same scale
A little bit of R
R
•
•
•
A statistical package
You can download the package from http://www.r-project.org/
Or
•
A good introduction at http://cran.r-project.org/doc/manuals/R-intro.pdf
Vectors
• A numeric vector is like a list of numbers in Python
• Index starts from 1
Command
What it does
x <- c(10,12,30,4,5)
Create a vector called x consisting of 10,12,30, 4, 5
x
Print out the contents of x
x[2:4]
Return 2nd to 4th entry in the vector
x[-3]
Return all entries in the vector except the 3rd entry
x[x<10]
Return all entries whose value is less than 10
Example commands for a vector
Command
What it does
length(x)
Number of values in x
mean(x)
Calculate the mean of x
median(x)
Calculate the median value of x
sd(x)
Calculate the standard deviation of x
var(x)
Calculate the variance of x
min(x)
Identify the minimum value in x
max(x)
Identify the maximum value in x
summary(x)
Summarize descriptive statistics of x
Data frames
• We often summarize our data as a table
– Each row is an observation characterized in terms of a
number of variables
– Each column lists values pertaining to a variable
• A data frame in R is like columns of vectors,
where each column can be labeled
>
>
>
>
>
a <- c(1,2,3,4)
b <- c(10,20,30,40)
c <- data.frame(v1=a,v2=b)
c$a
c$b
read.table()
• Read a file in table format and create a data
frame from it
– Specify the character that separates the fields
– e.g. sep=‘\t’
– Specify whether the file begins with headers
– e.g. header=TRUE
> v1<-read.table(‘/home/ling115/r/v1.txt’,sep=“\t”,header=TRUE)
> v2<-read.table(‘/home/ling115/r/v2.txt’,sep=“\t”,header=TRUE)
Correlation
• Let’s see how well the formants measured by two
students (v1 and v2) correlate
• v1$F1 refers to F1 values extracted by v1
• v2$F1 refers to F1 values extracted by v2
> cor(v1$F1,v2$F1)
> cor.test(v1$F1,v2$F1)
> cor.test(v1$F1,v2$F1,method=“spearman”)
• Likewise for F2