Transcript Check for normality – histogram

Post hoc tests
• F-test in ANOVA is the so-called omnibus test. It tests the
means globally. It says nothing about which particular
means are different.
• post hoc tests, multiple comparison tests.
• Tukey Honestly Significant Differences
>TukeyHSD(fit) #where fit comes from aov()
ANOVA assumptions
• normality – all samples are from normal distribution
• homogeneity of variance (homoscedasticity) – variances
are equal
• independence of observations – the results found in one
sample won't affect others
• Most influencial is the independence assumption.
Otherwise, ANOVA is relatively robust.
• We can sometimes violate
• normality – large sample size
• variance homogeneity – equal sample sizes + the ratio of any two
variances does not exceed four
Check for normality – histogram
Check for normality – QQ-plot
qqnorm(rivers)
qqline(rivers)
Check for normality – tests
• The graphical methods for checking data normality still
leave much to your own interpretation. If you show any of
these plots to ten different statisticians, you can get ten
different answers.
• H0: Data follow a normal distribution.
• Shapiro-Wilk test
• shapiro.test(rivers):
Shapiro-Wilk normality test
data: rivers
W = 0.6666, p-value < 2.2e-16
Homoscedasticity
http://blog.minitab.com/blog/statistics-and-quality-data-analysis/dont-be-a-victim-of-statistical-hippopotomonstrosesquipedaliophobia
Tests for homoscedasticity
• 𝐻0 : 𝜎1 = 𝜎2
• F-test of equality of variances (Hartley's test), 𝐹 𝑛𝐿 −
Nonparametric statistics
• Small samples from considerably non-normal
distributions.
• non-parametric tests
• No assumption about the shape of the distribution.
• No assumption about the parameters of the distribution (thus they
are called non-parametric).
• Simple to do, however their theory is extremely
complicated. Of course, we won't cover it at all.
• However, they are less accurate than their parametric
counterparts.
• So if your data fullfill the assumptions about normality, use
paramatric tests (t-test, F-test).
Nonparametric tests
• If the normality assumption of the t-test/ANOVA is
violated, and the sample sizes are too small, then their
nonparametric alternative should be used.
• The nonparametric alternative of t-test is Wilcoxon test.
> wilcox.test()
• The nonparametric alternative of ANOVA is Kruskal-
Wallis test
> kruskal.test()
ANOVA kinds
• one-way ANOVA (analýza rozptylu při jednoduchém
třídění, jednofaktorová ANOVA)
> aov(beer_brands$Price~beer_brands$Brand)
dependent variable
independent variable
• two-way ANOVA (analýza rozptylu dvojného třídění,
dvoufaktorová ANOVA)
• Example: engagement ratio, measure two educational methods
(with and without song) for men and women independently
> aov(engagement~method+sex)
• interactions between factors
CORRELATION
Introduction
• Up to this point we've been working with only one
variable.
• Now we are going to focus at two variables.
• Two variables that are probably related. Can you think of
some examples?
• weight and height
• time spent studying and your grade
• temperature outside and ankle injuries
Car data
Miles on a car
Value of the car
60 000
$12 000
80 000
$10 000
90 000
$9 000
100 000
$7 500
120 000
$6 000
• x – predictor, explanatory, independent variable
• y – outcome, response, dependent variable
Car data
Miles on a car
Value of the car
60 000
$12 000
80 000
$10 000
90 000
$9 000
100 000
$7 500
120 000
$6 000
• How may we show these variables have a relationship?
• Tell me some of yours ideas.
• scatterplot
Scatterplot
Stronger relationship?
Correlation
• Relation between two variables = correlation
• strong relationship = strong correlation, high correlation
Match these
strong positive
strong negative
weak positive
weak negative
Correlation coefficient
• r (Pearson's r) - a number that quantifies the relationship.
𝑟 = 𝑟𝑥𝑦
1
=
𝑛−1
𝑥𝑖 − 𝑥 𝑦𝑖 − 𝑦
𝑠𝑋 𝑠𝑌
+1
-1
+0.14
+0.93
-0.73
Coefficient of determination
• Coefficient of determination - 𝑟 2 is the percentage of
variation in Y explained by the variation in X.
• Percentage of variance in one variable that is accounted for by the
variance in the other variable.
r2 = 0
r2 = 0.25
r2 = 0.81
from http://www.sagepub.com/upm-data/11894_Chapter_5.pdf
Crickets
• Find a cricket, count the number of its chirps in 15
seconds, add 37, you have just approximated the outside
temperature in degrees Fahrenheit.
• National Service Weather Forecast Office:
http://www.srh.noaa.gov/epz/?n=wxcalc_cricketconvert
chirps in 15 sec temperature chirps in 15 sec temperature
18
57
27
68
20
60
30
71
21
64
34
74
23
65
39
77
Hypothesis testing
• Even when two variables describing the sample of data
may seem they have a relationship, this could just be due
to the chance. The situation in population may be
different.
• 𝑟 … sample corr. coeff., 𝜌 … population corr. coeff.
• How hypotheses will look like?
𝐻0 : 𝑟 = 0
𝐻𝐴 : 𝑟 < 0
𝑟>0
𝑟≠0
𝐻0 : 𝜌 = 0
𝐻𝐴 : 𝜌 < 0
𝜌>0
𝜌≠0
𝐻0 : 𝑟 < 0
𝑟>0
𝑟≠0
𝐻𝐴 : 𝑟 = 0
𝐻0 : 𝑟 < 0
𝑟>0
𝑟≠0
𝐻𝐴 : 𝑟 = 0
A
B
C
D
Hypothesis testing
𝑡=
𝑟 𝑛−2
1
− 𝑟2
with 𝑑𝑓 = 𝑛 − 2
• test statistic has a t-distribution
• Example: we measure the relationship between two
variables, we have 25 participants, we get t = 2.71. Is
there a significant relationship between X and Y?
• 𝛼 = 0.05, non-directonal test, 𝑡𝑐𝑟𝑖𝑡 = 2.069
Correlation vs. causation
• causation – one variable causes another to happen
• For example, the fact that it is raining causes people to take their
umbrellas .
• correlation – just means there is a relationship
• For example, do happy people have more friends? Are they just
happy because they have more friends? Or they act a certain way
which causes them to have more friends.
Correlation vs. causation
• There is a strong relationship
between the ice cream
consumption and the crime rate.
• How could this be true?
• The two variables must have
something in common with one
another. It must be something
that relates to both level of ice
cream consumption and level of
crime rate. Can you guess what
that is?
• Outside temperature.
from causeweb.org
Correlation vs. causation
• Outside temperature is a variable we did not realize to
control.
• Such variable is called third variable, confounding
variable, lurking variable.
• The methodologies of scientific studies therefore need to
control for these factors to avoid a 'false positive‘
conclusion that the dependent variables are in a causal
relationship with the independent variable.
Correlation vs. causation
• If you stop selling ice cream, does the crime rate drop?
What do you think?
• That’s because correlation expresses the association
between two or more variables; it has nothing to do with
causality.
• In other words, just because level of ice cream
consumption and crime rate increase/descrease together
does not mean that a change in one necessarily results in
a change in the other.
• You can’t interpret associations as being causal.
http://xkcd.com/552/
Correlation and regression analysis
• Correlation analysis investigates the relationships
between variables using graphs or correlation coefficients.
• Regression analysis answers the questions like: which
relationship exists between variables X and Y (linear,
quadratic ,….), is it possible to predict Y using X, and with
what error?
Simple linear regression
• also single linear regression
• one y (dependent variable,
(jednoduchá lineární regrese)
závisle proměnná),
one x
(independent variable, nezávisle proměnná)
• 𝑦 = 𝑎 + 𝑏𝑥
• 𝑎 – y-intercept (constant), 𝑏 – slope
• 𝑦 is estimated value, so to distinguish it from the actual
value 𝑦 corresponding to the given 𝑥 statisticans use 𝑦
Data set
• Students in
higher grades
carry more
textbooks.
• Weight of the
textbooks
depends on the
weight of the
student.
strong positive correlation, r = 0.926
outlier
from Intermediate Statistics for Dummies
Build a model
• Find a straight line y = a + bx
Interpretation
• y-intercept (3.69 in our case)
• it may or may not have a practical meaning
• Does it fall within actual values in the data set? If yes, it is a clue it
may have a practical meaning.
• Does it fall within negative territory where negative y-value are not
possible? (e.g. weights can’t be negative)
• Does a value x = 0 have practical meaning (student weighting 0)?
• However, even if it has no meaning, it may be necessary (i.e.
significantly different from zero)!
• slope
• change in y due to one-unit increase in x (i.e. if student’s
weight increases by 1 pound, its textbook’s weight increases
by 0.113 pounds)
• now you can use regression line to estimate y value
for new x
Regression model conditions
• After building a regression mode you need to check if the
required conditions are met.
• What are these conditions?
• The y’s have normal distribution for each value of x.
• The y’s have constant spread (standard deviation) for each value of
x.
Normal y’s for every x
• For any value of x, the population of possible y-values
must have a normal distribution.
from Intermediate Statistics for Dummies
Homoscedasticity condition
As you move from the left to the right on the x-axis, the
spread around y-values remain the same.
source: wikipedia.org
Residuals
• To check the normality of y-values you need to measure
how far off your predictions were from the actual data, and
to explore these errors.
• residual (residuum, reziduální hodnota predikce)
𝑒 =𝑦−𝑦
actual value
residual
predicted value
from Intermediate Statistics for Dummies
Residuals
• The residuals are data just like any other, so you can find
their mean (which is zero!!) and their standard deviation.
• Residuals can be standardized, i.e. converted to the Zscore so you see where they fall on the standard normal
distribution.
• Plotting residuals on the graph – residual plots.
normality of residuals
homoscedasticity
residuals independence
Using r2 to measure model fit
• r2 measures what percentage of the variability in y is
explained by the model.
• The y-values of the data you collect have a great deal of
variability.
• You look for another variable (x) that helps to explain the
variability in the y-values.
• After you put x into the model and you find it’s highly
correlated with y, you want to find out how well this model
did at explaining why the values of y are different.
Interpreting r2
• high r2 (80-90% is extremely high, 70% is fairly high)
• A high percentage of variability means that the line fits well because there is
not much left to explain about the value of y other than using x and its
relationship to y.
• small r2 (0-30%)
• The model containing x doesn’t help much in explaining the difference in the yvalues
• The model would not fit well. You need another variable to explain y other than
the one you already tried.
• middle r2 (30-70%)
• x does help somewhat in explaining y, but it doesn’t do the job well enough on
its own.
• Add one or more variables to the model to help explain y more fully as a group.
• Textbook example: r = 0.93, r2 = 0.8649. Approximately 86% of
variability you find in textbook weights for is explained by the average
student weight. Fairly good model.
Multiple regression
• Two (or more) variables are better than one.
• y = b0 + b1x1 + b2x2 + … + bkxk
• Steps in the analysis
• Check the relationships between each x variable and y (using
scatterplots and correlations) and use the results to eliminate those
x variables that aren’t strongly related to y.
• Look at possible relationships between the x variables themselves
to make sure you aren’t being redundant (in statistical terms, you’re
trying to avoid the problem of multicolinearity). If two x variables
relate to y the same way, you don’t need both in the model.
• Use selected x variables in a multiple regression analysis to find
the best-fitting model for your data.
• Use the best-fitting model to predict y for given x-values.
Data set
Relate plasma TV sales with
two types of advertisement .
from Intermediate Statistics for Dummies
Pinpoint Possible Relationships
Scatterplot
from Intermediate Statistics for Dummies
Correlations
• All possible correlations
• TV vs. Sales
• Newspaper vs. Sales
• Newspaper vs. TV
from Intermediate Statistics for Dummies
• Is correlation coefficient ρ statistically significant?
• What is the null hypothesis?
• Ho: ρ = 0, Ha: ρ ≠ 0
• Investigate p-values.
• If p-value is smaller than α (typically 0.05), reject Ho.
from Intermediate Statistics for Dummies
Checking for Multicolinearity
• Look at the relationship between the x variables
themselves and check for redundancy.
• Multicolinearity – two x variables are highly correlated.
• If two x variables are significantly correlated, include only one.
• If you include both, the computer won’t know what numbers to give
as coefficients for each of the two variables, because they share
their contribution to determining the value of y.
• Multicolinearity can really mess up the model-fitting.
from Intermediate Statistics for Dummies
Find the best fitting model
from Intermediate Statistics for Dummies
• The interpretation of coefficients are a little more
complicated then in simple linear regression.
• The coefficient of an x variable in a multiple regression model is the
amount by which y changes if that x variable increases by one and
the values of all other x variables in the model don’t change.
• Plasma TV sales increases by 0.162 million dollars when TV ad
spending increases by $1,000 and spending on newspaper ads
doesn’t change.
Testing the Coefficients
• Determine whether you have the right x variables in your
model.
• Test Ho: Coef = 0, Ha: Coef ≠ 0
Extrapolation: no-no
• Do not estimate y for values of x outside their range!
• There is no guarantee that the relationship you found
follows the same model for distant values of predictors.
Checking the fit of model
The residuals have a normal distribution with mean
zero.
2. The residuals have the same variance for each fitted
(predicted) value of y.
3. The residuals are independent (don’t affect each other).
1.
normality of residuals
homoscedasticity
residuals independence
from Intermediate Statistics for Dummies
Adjusted R2
• How well the regression line approximates the real data points is
measured by the coefficient of determination
R 2 = r 2.
• It tells you how much the variability in the y-value is explained by the
model.
• R2 increases as we increase the number of variables in the model.
• As such, R2 alone cannot be used as a meaningful comparison of
models with different numbers of independent variables – adjusted
R2.
n 1
R  1  1  R 
n  p 1
2
2