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== $t$ Tests ==
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== Family of $t$ Tests ==
$t$-tests is a family of [[Hypothesis Testing|Statistical tests]] that use $t$-statistics (those values come from the [[t Distribution|$t$-distribution]]) to calculate $p$-values
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$t$-tests is a family of [[Hypothesis Testing|Statistical tests]] that use $t$-statistics  
 +
* critical values come from the [[t Distribution|$t$-distribution]] - used for calculating $p$-values
  
  
 +
== $t$ Tests ==
 
The following tests are $t$-tests:
 
The following tests are $t$-tests:
* one-sample $t$-test - for comparing the mean of a sample against some given mean
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* [[One-Sample t-test|One-Sample $t$-test]] - for comparing the mean of a sample against some given mean
* two-sample $t$-test - for comparing the means of two samples
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* [[Two-Sample t-test|Two-Sample $t$-test]] - for comparing the means of two samples
* paired $t$-test - for Matching Pairs setup
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* [[Paired t-test|Paired $t$-test]] - for Matching Pairs setup
* pairwise $t$-test - for comparing the means of more than two samples
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* [[Pairwise t-test|Pairwise $t$-test]] - for comparing the means of more than two samples
  
  
== When To Use ==
 
 
=== Assumptions ===
 
=== Assumptions ===
 
Assumptions for $t$ tests are similar to the assumptions of the [[z-tests|$z$-tests]]
 
Assumptions for $t$ tests are similar to the assumptions of the [[z-tests|$z$-tests]]
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=== vs [[z-tests|$z$-tests]] ===
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=== $t$-tests vs [[z-tests|$z$-tests]] ===
 
Sample Size
 
Sample Size
 
* the sample size can be smaller than for $z$-tests  
 
* the sample size can be smaller than for $z$-tests  
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* the tails are thicker than for $N(0,1)$ and observations are more likely to fall within 2$\sigma$ from the mean
 
* the tails are thicker than for $N(0,1)$ and observations are more likely to fall within 2$\sigma$ from the mean
 
* this is exactly the correction we need to account for poorly estimated [[Standard Error]] when the sample size is not big
 
* this is exactly the correction we need to account for poorly estimated [[Standard Error]] when the sample size is not big
 
 
 
== One-Sample t-test ==
 
This is a test for one variable
 
* it's used mainly to calculate a [[Confidence Intervals|Confidence Interval]] for the true mean $\mu$
 
* the null value for $H_0$ might come from other research or from your knowledge
 
 
Parameters:
 
* $\text{df} = n - 1$ with $n$ being the sample size
 
 
 
=== Example 1 ===
 
* Sample: $n = 60, \bar{X} = 7.177, s = 2.948$
 
* True mean $\mu$ is unknown
 
 
Let's run a test:
 
* $H_0: \mu = 0, H_A: \mu > 0$ (this is one-sided test)
 
* Under $H_0$, we know that $\cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \approx t_{n - 1}$
 
* Observed: $\bar{X} - \mu = 7.177 - 0 = 7.177$
 
* How plausible is the observed value under $H_0$?
 
 
 
The probability of observing this value is
 
* $P(\bar{X} - \mu \geqslant 7.177) = $
 
** $P\left(\cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \geqslant \cfrac{7.177}{\sqrt{s^2 / n}}\right) \approx$
 
** $P\left(t_{59} \geqslant \cfrac{7.177}{\sqrt{2.948^2 / 60}}\right) \approx$
 
** $P(t_{59} \geqslant 18.86) \approx 1 / 10^{26}$
 
 
Extremely small! So we reject $H_0$ and conclude that $\mu > 0$
 
 
 
 
=== Example 2 ===
 
* Sample $n = 400, \bar{X} = -14.15, s = 14.13$
 
* Test: $H_0: \mu = 0$  vs $H_A: \mu \neq 0$ (this is a 2-sided)
 
 
We know that
 
* $\cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \approx t_{n - 1} = t_{399}$
 
 
 
$p$-value:
 
* $P( | \bar{X} - \mu | \geqslant | -14.15 - 0 |) = $
 
** $P\left( \left| \cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \right| \geqslant \cfrac{14.15}{\sqrt{14.13^2 / 400}}\right) \approx $
 
** $P( | t_{399} | \geqslant 20.03 ) =$
 
** $2 \cdot P( t_{399} \leqslant -20.03) \approx$
 
** $1 / 3.5 \cdot 10^{64}$
 
 
 
Extremely small! Reject the $H_0$ and conclude that $\mu \neq 0$
 
 
 
=== R code ===
 
Our test statistic is $T = \cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}}$.
 
 
<pre>
 
xbar = mean(ch)
 
s2 = var(ch)
 
n = length(ch)
 
mu = 0
 
 
t = (xbar - mu) / sqrt(s2 / n) // 18.856
 
pt(t, df=n-1, lower.tail=F) // 5.84E-24
 
// so we reject
 
</pre>
 
 
 
 
 
== Paired t-test ==
 
=== Paired Data ===
 
Two set of observations are ''paired'' if each observation in one set has exactly one corresponding observation is another set.
 
 
 
Examples:
 
* pre- and post-test scores on the same person
 
* measures in pairs at the same time or place
 
* outcome with or without a treatment - on same subject (cross-over study)
 
 
 
=== [[R]] Function ===
 
<pre>
 
library(openintro)
 
data(textbooks)
 
t.test(textbooks$diff, mu=x.bar.nul, alternative='two.sided')
 
</pre>
 
 
or
 
 
<pre>
 
t.test(textbooks$uclaNew, textbooks$amazNew, paired=T,
 
      alternative='two.sided')
 
</pre>
 
 
 
 
=== Example: Bookstore vs Amazon ===
 
* two samples: local bookshop and amazon
 
* $\mu_\text{dif} = \mu_l - \mu_a$ - the mean of difference in the price
 
 
Test
 
* $H_0: \mu_\text{dif} = 0$ - there's no difference in the price
 
* $H_A: \mu_\text{dif} \ne 0$ - there's some difference
 
 
Calculations
 
* $\bar{x}_\text{dif} = 12.76$
 
* Standard Error: $\text{se}_{\bar{x}_\text{dif}} = \cfrac{s_\text{dif}}{\sqrt{n_\text{dif}}} = 1.67$
 
* $T = \cfrac{\bar{x}_\text{dif}}{\text{se}_{\bar{x}_\text{dif}}} = \cfrac{12.76}{1.67} = 7.59$
 
* $p = 6 \cdot 10^{-11}$, less than $\alpha = 0.05$, so we reject $H_0$
 
 
 
<pre>
 
library(openintro)
 
data(textbooks)
 
 
hist(textbooks$diff, col='yellow')
 
 
n = length(textbooks$diff)
 
s = sd(textbooks$diff)
 
se = s / sqrt(n)
 
 
x.bar.nul = 0
 
x.bar.dif = mean(textbooks$diff)
 
 
t = (x.bar.dif - x.bar.nul) / se
 
t
 
p = pt(t, df=n-1, lower.tail=F) * 2
 
p
 
</pre>
 
 
or
 
 
<pre>
 
t.test(textbooks$diff, mu=x.bar.nul, alternative='two.sided')
 
</pre>
 
 
 
=== Example 2 ===
 
Let $\mu_d = \mu_0 - \mu_1$ be the difference between two methods
 
 
Our test:
 
* $H_0: \mu_d = 0, H_A: \mu_d \neq 0$
 
 
Say, we have:
 
* $\bar{X}_d = 6.854$
 
* $s_d = 11.056$
 
* $n = 398$
 
 
Test statistics:
 
* $\cfrac{\bar{X}_d - 0}{s_d / \sqrt{n}} = \cfrac{6.854}{11.056 / \sqrt{398}} \approx 12.37$
 
 
Then we compare it with $t_{397}$
 
* $p$-value is $2.9 \cdot 10^{29}$
 
 
And we conclude that the difference between the two methods is not 0
 
 
 
 
== Two-Sample t-test ==
 
This variation of $t$-test is used when we want to compare the means of two different samples
 
* suppose that we have two samples $a$ and $b$ of sizes $n_a$ and $n_b$ resp.
 
* we're interested in inferring something about $\mu_a - \mu_b$
 
* [[Point Estimate]] in this case is $\bar{x}_a - \bar{x}_b$
 
* [[Standard Error]] is $\text{SE}_{\bar{x}_a - \bar{x}_b} = \sqrt{\text{SE}_a + \text{SE}_b } = \sqrt{ s^2_a / n_a + s^2_b / n_b}$
 
** because $\text{SE}^2_{\bar{x}_a - \bar{x}_b} = \text{var}[\bar{x}_a - \bar{x}_b] = \text{var}[x_a] + \text{var}[x_b] = \text{SE}^2_a + \text{SE}^2_b$
 
 
 
The test is of the following form
 
* $H_0: \mu_a = \mu_b$, or $H_0: \mu_a - \mu_b = 0$
 
* $H_A: \mu_a \neq \mu_b$ or $H_A: \mu_a - \mu_b \neq 0$ (two-sided, can also be $<$ or $>$)
 
 
 
So, test statistics:
 
* $T = \cfrac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)}{\sqrt{s_1^2 / n_1 + s_2^2 / n_2}}$
 
* $T \approx t_{\text{df}}$
 
* $\text{df}$ depends on a few things, discussed below
 
 
 
=== Welch-Satterthwaite Approximation ===
 
What is $\text{df}$ there?
 
* Welch-Satterthwaite Approximation for df is
 
* $\text{df} = \cfrac{( s_1^2 / n_1 + s_2^2 / n_2 )^2 }{ \frac{(s_1^2 / n_1)^2 }{n_1 - 1} + \frac{(s_2^2 / n_2)^2 }{n_2 - 1} }$
 
 
This can be a non-integer value, but that's fine
 
 
 
=== Pooled Variance Estimation ===
 
* Can we "pool" the samples?
 
* Yes, but only under assumption that $\sigma_1^2 = \sigma_2^2$ (in other words, we assume that the variances are equal)
 
 
We can replace $s_1^2$ and $s_2^2$ by the ''pooled variance'':
 
* $s^2 = \cfrac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2 }{ (n_1 - 1) + (n_2 - 1)}$
 
* and $\text{df} = (n_1 - 1) + (n_2 - 1) = n_1 + n_2 - 2$
 
 
 
 
=== Example 1 ===
 
* males: $n_1 = 281 $
 
* females: $n_2 = 199$
 
* $\bar{X}_1 = -12.9, s_1^2 = 181.5$
 
* $\bar{X}_2 = -17.1, s_2^2 = 231.5$
 
* $\bar{X}_1 - \bar{X}_2 = -12.9 + 17.1 = 4.2$
 
 
We then calculate
 
* $\text{df} = 200.09$
 
* so $T_{0.025, 200.09} = 1.97$
 
 
 
We have the following test
 
* $H_0: \mu_1 = \mu_2, H_A: \mu_1 \neq \mu_2$
 
* and $\bar{X}_1 - \bar{X}_2 = 4.2$
 
 
 
$p$-value:
 
* $P(| \bar{X}_1 - \bar{X}_2 |  \geqslant  4.2 ) = $
 
* $P \left( \left| \cfrac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)}{\sqrt{s_1^2 / n_1 + s_2^2 / n_2}} \right|  \geqslant  \cfrac{4.2}{\sqrt{s_1^2 / n_1 + s_2^2 / n_2}} \right) \approx $
 
* $P\left( |t_\text{df} |  \geqslant \cfrac{4.2}{\sqrt{181.5 / 281 + 231 / 119}} \right) = 0.0097$
 
 
pretty small, so we reject the $H_0$.
 
 
 
=== Example 2 ===
 
Life expectancy in E.Asia and Pacific vs S.Asia
 
* EA&P: $n_1: 30, \bar{X}_1 = 73.1, s_1^2 = 38.7$
 
* SA: $n_2: 8, \bar{X} = 67.0, s_2^2 = 72.5$
 
* $\bar{X}_1 - \bar{X}_2 = 73.1 - 67.0 = 6.1$
 
 
 
We then calculate
 
* $\text{df} = 9.09$ by Welch-Satterthwaite Approximation
 
* $T_{0.025, 0.09} = 2.26$
 
 
 
Our test:
 
* $H_0: \mu_0 = \mu_1, H_A: \mu_0 \neq \mu_1$
 
 
$p$-value:
 
* $P(| \bar{X}_1 - \bar{X}_2 |  \geqslant  6.1 ) = $
 
* $P \left( \left| \cfrac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)}{\sqrt{s_1^2 / n_1 + s_2^2 / n_2}} \right|  \geqslant  \cfrac{6.1}{\sqrt{s_1^2 / n_1 + s_2^2 / n_2}} \right) \approx $
 
* $P ( |t_\text{df} |  \geqslant 1.90 ) \approx 0.09$
 
 
Not so small - we can't reject the $H_0$, it might be true that $\mu_0 = \mu_1$
 
 
 
=== R (Means) ===
 
<pre>
 
male = skeletons[sex == '1', 6]
 
female = skeletons[sex == '2', 6]
 
 
# critical value
 
qt(0.025, df=200.9, lower.tail=F)
 
</pre>
 
 
or
 
 
<pre>
 
t.test(male, female, mu=0, conf.level=0.95, alternative='two.sided')
 
</pre>
 
 
 
 
== Pairwise t-test ==
 
* we have $n$ groups, $n > 2$
 
* we conduct a series of Two-Sample t-tests to find out which groups are different
 
* e.g. in post-[[ANOVA]] analysis
 
 
 
=== Controlling [[Family-Wise Error Rate]] ===
 
It's important to modify $\alpha$ to avoid [[Type I Errors]]
 
* when we run many tests, it's inevitable that we make them just by chance
 
 
E.g. use [[Bonferroni Correction]]
 
* use modified confidence level $\alpha^* = \alpha \cdot \cfrac{1}{K}$
 
* where for $k$ groups $K= \cfrac{k \cdot (k - 1)}{2}$
 
  
  
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* http://projectile.sv.cmu.edu/research/public/talks/t-test.htm
 
* http://projectile.sv.cmu.edu/research/public/talks/t-test.htm
  
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[[Category:T-Test]]
 
[[Category:Statistics]]
 
[[Category:Statistics]]
 
[[Category:Statistical Tests]]
 
[[Category:Statistical Tests]]
 
[[Category:R]]
 
[[Category:R]]

Revision as of 18:32, 23 November 2015

Family of $t$ Tests

$t$-tests is a family of Statistical tests that use $t$-statistics

  • critical values come from the $t$-distribution - used for calculating $p$-values


$t$ Tests

The following tests are $t$-tests:


Assumptions

Assumptions for $t$ tests are similar to the assumptions of the $z$-tests

  • Observations are independent (if less than 10% of population is sampled, then we can make sure it's satisfied)
  • Sample size is sufficiently large so C.L.T. holds
  • Moderate skew, few outliers (not too extreme)


$t$-tests vs $z$-tests

Sample Size

  • the sample size can be smaller than for $z$-tests
  • so it can be smaller than 30 - after 30 we can safely use $z$-tests with almost the same outcomes

$t$-distribution:

  • the tails are thicker than for $N(0,1)$ and observations are more likely to fall within 2$\sigma$ from the mean
  • this is exactly the correction we need to account for poorly estimated Standard Error when the sample size is not big


Sources

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