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97 Cards in this Set
- Front
- Back
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response variable
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y
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explanatory variable
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x
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population regression line
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uY=Bo+B1x
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sample/model regression line equation
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yhat=bo+b1x
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bo
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intercept
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b1
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slope
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subpopulation
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the normal curve around each different value of x
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UY
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the actual average response for y given a value of x
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sdeviations of x
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are the same for all vlaue of x because , x values are part of a subpop on a normal curve.
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ei
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residual for a given x , obs-predicted or yi-yhati
or yi - (model regression equation, which gives yahti) |
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s^2
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is variance (thus s is SD)
s2= sum(yi-yhati)^2/n-2 you will not have to calculate this as it is given in summary fit |
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in summary fit the "residual standard error"
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is the same as "s" or standard deviation
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when we estimate the standard error using "s" we move to the t dist. how many df does the t dist have
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n-2
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list 4 different types of confidence intervals for t distributions in simple regression
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b1+-t*seb1
bo+-t*sebo mean resp Uy+-t*seuy (formula and r code ) prediction internval yhat+-t*seyhat |
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critical t in r
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qt(.95,n-2)
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regular (not r formula) for se of mean response
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s*sqrt ( 1/n) + xstar - xbar ^2 / sum (xi-xbar)^2
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c level predication interval se formula (interval for yhat)
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s*sqrt( 1 + 1/n + ((((( xstar - xbar ^2 / sum (xi-xbar)^2 )))))))
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why does a small p value for a large n not mean there is a strong corr.
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because , a large n helps us estimate the true mean, but it does not reduce scatter.
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formula for sse
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sum ( yi-yhat) ^2
observed deviation from model |
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formula for ssm
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sum ( yhat-ybar) ^2
model deviation from actual |
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sst
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sum ( yi - ybar) ^2
total deviation from obs to actual |
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if Ho: b1=0 were true the variation of y on x would be the same as
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s2 or MST
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t dist. dfm
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1
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t dist. dfe
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n-2
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t dist dft
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n-1
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mean square
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ss/df
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list forumas for msm, mse, mst
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ssm/dfm
sse/dfe sst/dft |
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r2 =
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ssm/sst
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f = (using anova variables)
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msm/mse
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t^2
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f
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how to get t
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b1/seb1 (given in summary fit)
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rho ( p ) is what
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population correlation (r is sample correlation)
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if rho p=o
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then there is no linear relation, x and y are independent.
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sig test for rho
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is found using the t value just like for b1, but watch out as sometimes we want to only do a one sided test for rho.
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in multiple linear regression p is
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number of explanatory variables
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in multiple linear regression I (capital i or j ) is
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a specific explanatory varible
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in multiple linear regression i is
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one case for one explanatory variable
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in multiple linear regression n is
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n is the total number of values i
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in multiple linear regression (one way anova) N is
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the total number of n's - the sum of n's for each explanatory variable.
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multiple regression format
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multiple regression
yhat=bo+ b1xij...... |
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how to get t
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b1/seb1 (given in summary fit)
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rho ( p ) is what
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population correlation (r is sample correlation)
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if rho p=o
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then there is no linear relation, x and y are independent.
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sig test for rho
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is found using the t value just like for b1, but watch out as sometimes we want to only do a one sided test for rho.
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in multiple linear regression p is
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number of explanatory variables
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in multiple linear regression I (capital i or j ) is
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a specific explanatory varible
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in multiple linear regression i is
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one case for one explanatory variable
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in multiple linear regression n is
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n is the total number of values i
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in multiple linear regression (one way anova) N is
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the total number of n's - the sum of n's for each explanatory variable.
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multiple regression format
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multiple regression
yhat=bo+ b1xij...... |
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in multiple linear regression - explain the total of I j p i n and N
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sum of j/I = p
sume of i = n sum of n = N remember N observations and P explanatory variables |
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sig test for bo in multiple regression
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same bj/sej given in summary fit but
pt(t value, with df n-p-1) not just n-2 |
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hyp test for multiple regression
hypotheses |
Ho: all slopes = o
Ha: at least one explanatory variable not = o |
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to execute a hyp test for multiple regression use the ___ test stat
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f
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in multiple regression DFM is
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p
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in multiple regression DFE is
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n-p-1
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in multiple regression DFT is
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n-1 ( like in simple regression)
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how to take f stat
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msm/mse
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sqrt( R ^ 2 )
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the multiple correlation coefficient - correlation between yi and yhati
(is is a corr between yaht and y) |
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why does a low p value not imply a large R^2
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a statistically significant result is much different than a large effect of x on y .
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factor
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categorical explanatory variable
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again how to find f
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msm/mse
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t^2 is
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f
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sig test for rho p involves
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Ho: p not = 0 use t test (if p=0 then the relation is not linear and x , y are independent) (sometimes this test will be one sided)
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how to find p value for t
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2*(1-pt(tstat, df n-2 for simple and n-p-1 when testing 1 P out of a multiple regression).
(when testing multiple P's use the F test) |
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what is a one way anova
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compares several population means
- a single quantitative response variable compared to several populations. |
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2 sample t test formula (both sample sizes same n) for an individual mean1
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t ij = (sqrt(n/2) * x1bar-x2bar )/ sp
where sp =sqrt(mse) |
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2 sample t test formula (both sample sizes same n) for all means
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t ij = (sqrt(n/2) * x1bar-x2bar )/ sp
where sp =sqrt(mse) take t and make it t^2 for f |
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R code to run an F test for 1 way anova
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juice<-data.frame(vector,vector)
juice<-stack(juice) names(juice) oneway.test(values~ind,data=juice,var.equal=T) fit<-anova(lm(values~ind,data=juice) |
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t test for one way anova
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t.test(vector,vector,alternative=c("two.sided"),mu=0,paired=FALSE,var.equal=TRUE,conf.level=.95)
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hypotheses in one way anova tests
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ho: all means =
ha: not all means are = |
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in order to assume equal SD's in one way anova
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the largest sd must not be more than 100% the smallest
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incase sqrt(mse is too much trouble to find sp, the manual way to find sp^2 is
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(n1-1)*s1^2 + (n2-1)*s2^2....../(n1-1)+(n2-1).....
****s can be found in r by using the command "sd" sd(vector) |
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I
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number of populations to be compared
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one way anova hypotheses
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Ho: u1=u2=u3.....
Ha: not all Ui are = |
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one way anova DFT
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n-1
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one way anova DFG
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group
I-1 |
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one way anova DFE
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N-I
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one way anova R^2
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SSG/SST
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to find the value of a residual given an x or y
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plug it into the model and take the difference between what the model predicts and what was given
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r code for getting to summary fit
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set x and y vectors
fit<-lm(y~x) can be +x2+x3... summary(fit) |
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how to find p value for f
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pf(f, DFM, DFE)
dfm and dfe vary depending on weather this is an f test of multiple regression or a 1 way ANOVA test. |
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how to get anova chart in r
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anova(fit)
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in a hyp test conclusion:
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re state hypotheses
re state variables |
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how to conclude, if we reject null in a multiple regression F test
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if we reject null then the , there is a low chance all the variables produce a slope of zero, so at least one x influces y and model is valid
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1-r^2
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unexplained x on y
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if p variables are greater than N observations
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there could be alot of conflict
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in 1 way ANOVA DFG is
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I-1
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in 1 way ANOVA DFE is
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N-I
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in 1 way ANOVA DFT is
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N-1
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in 1 way ANOVA MSG is
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SSG/DFG
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in 1 way ANOVA MSE is
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SSE/DFE
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in 1 way ANOVA F is
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MSG/MSE
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as DF gets bigger for the same groups
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the resulting p values get small - bc less variability is allowed fo rit to be statistically signifigant.
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if asked "what are the df for the anova f stat'
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list both numerator and denomenator DFG and DFE , but default to DFE
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R code for SE of mean response
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vx<-var(x) and s is given in summary fit
se<-s*sqrt((1/length(x))+((xstar-xbar)^2/((length(x)-1)*vx |
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se for prediction interval yhat
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vx<-var(x) and s is given in summary fit
se<-s*sqrt(1+(1/length(x))+((xstar-xbar)^2/((length(x)-1)*vx))) |
