Lecture 7: T-Tests Flashcards
T-Tests
* A stastical ratio used to compare 2 group means
* tells us if the two groups of data are statistically different or is there a chance that the two groups are not statistically different from one another
X bar = the mean of the group
What is another name for an independent T-Test?
* What is it?
Unpaired T-Test
Means that the two groups of data being used are different (isnt the same group being tested twice
So in the picture below group 1 would be 4 different people and group 2 would be four different people
However, if it was the same group of people being tested twice (think anatomy score vs physiology score) than it would be considred a paired T-Test
* Because for each person there is a pair of data
A statistical ratio used to compare 2 group means dividuded by variability within groups
T-Test
Variability = the variation between scores within 1 group (so in group one its the variabition between 91, 100, 98, 100).
X bar = the mean for each group
A T Test would be the difference between means / Variability within groups
Does A t test take variance into consideration?
Yes, it looks at how variable each data set is
Homogeneity is assumed w/ in t-tests. What does that mean?
Variability in data is similar (you can’t have one group thats extremely variable and another group that has almost no variability)
Is the variability between these two data sets similar or different?
* Is it low or high vairability?
Similar
* The curves look the same
Low variability
* the curve is smooshed together, meaning all the data points fall close to the mean = decreased vairability
Which groups have a larger variability B or C?
C
* The wider it is = more variability
There is more variability within each data set in C because the curves are wider, meaning the points in each curve fall on average further from the mean than in B
NOTE In B each of the two graphs (orange and blue) have about the same width. Its the same thing for C (both orange and blue have about the same width). Meaning within B and within C a t-test would work well, beacuse for a t-test we want both graphs to have similar variable
* NOTE: I am not saying the variability between C and B is the same, im saying within each graph (B and C) the two data sets graphed have similar variability = same variabce = good for a t-test
This is what no variability within the groups looks like. If everyone got the same socre in each group you can see no deviation from the mean
* So we can obviously tell that there is a significiant difference between groups
Why would a t-test not be great here?
No t-test here because the variability between the two curves is very different. In T-Tests we want roughly equal variability between data sets
KNOW: When the variability is very different between curves theres a different kind of T-test we use
Here theres a small difference, the means are pretty close together
but theres a bigger difference in variance between the two. Which according the the equation below will lower our T value
T = Small difference / Large variance std deviation
* Small # over big number = small T
Theres tons of overlap = not a significant difference between the two groups = Small T
* So theres a small chance they’re actaully statistically different
Small variance between groups
Larger Difference in means
t = large difference/small variance std deviation = Big T value
Theres only a small amount of overlap, meaning theres a significant difference between the two (which is why we we got a small T)
A Large T or Small T value = A bigger difference between the two groups?
The bigger T is the bigger the difference between the two data sets
To do the T-Test we use the equation Difference between means / Variability within groups
Means: 20-10 = 10
Variance: 3-37?
10/34 = .29
T-Test yields a small value of .29
* which means theres not much staitstical difference
To find T we use the equaltion difference between means/Variability within groups
9-5/12-7? = .8
NOTE: this is a bigger T value = more statistical significant difference
* Makes sense, theres not much overlap
A t-test assumes similar variability between the two groups. Below you can see that the means are the exact same, however, the variance is not even close to the same between the two groups. This data set would not be a good canidate for a normal t-test
* A normal T-test assumes homogeneity (variation is similar)
This is a limiation of the t-test because it would assume the groups are the exact same, however, we can see that they clearly are not due to the variation
When a t-test is performed, a test of homogeneity of variance is performed
* meaning, they take the variance of one group, and comapre it to the variance of the other group and see if theres a stasticially difference between the two groups varibility
This is looking at an independent/unpaired T-Test (meaning there are two seperate groups of people, groups are considered independent, subjects are different in each group). I.e., treatment vs contorl group.
X bar = the mean of each group
Sxbar = standard error of the difference between the means = whats the difference in the variability (difference in variability of one group vs the other group)
* Also can just consider this the pooled variance (what the variance is overall)
This is still talking about the Independent T-Test from the slide prior
This is looking at two different groups. One group is using the newly developed splint while the other is using the standard splint.
The two measures that we are going to compare is the total chance in grip strength (so the two final measurs of each group)
n = # of people in each group
New splint = gained on average 10.1100 pounds of grip strength
Standard = 5.4500 pounds of grip strength gained
Mean difference between the groups = 4.66 (10.1100-5.4500)
* So we want to know if this is truely significant or just due to chance
We assume equal variance between were doing a T-test
we find that t = 2.718. This doesnt mean much, we don’t know if this is large enough to be significant
So in our splint example we are going to do a 2 tailed t test because its nondirectional. Meaning, we don’t know if the new splint group is going to get more strength gains or go down in strength gains. We just don’t know what that realtionship is going to look like
dont memorize these
she wants us to understand how the variance is pooled together
* Know its pooled between the two groups
above we found that the t-value was 2.718
* However, this doesnt have much meaning to use alone
* We need to compare the T-value to the critical value to find meaning
Critical value: A value set based on a pre-specified level of significance (a) (0.05 typically)
* Compare t to the critical value to determine significance
* Can be done for 1-tailed and 2 tailed t tests
One tailed t test vs 2 tailed t test
1 tailed t test - less common - have to prove without a shadow of a doubt it can only go 1 direction
* Purpose is to test for the possibility of the relationship in one direction
* The null hypothesis H0 states that there is no effect or difference, while the alterantive hypothesis states that there is a difference in a specific direction (greater or lass than)
* EX: H0: μ1 = μ2 (no difference) vs. H1: μ1 > μ2 (group 1 mean is greater than group 2 mean).
* Use case: Use a one-tailed test when you have a specific prediction about the direction of the effect and are sure its not going to go the other way
Two tailed T-Test
* Purpose: Tests for the possibility of the relationship in both directions
* Hypothesis: The null hypothesis H0 remains the same, but the alternative hypothesis states that there is a difference, without specifying a direction
* Example: H0: μ1 = μ2 (no difference) vs. H1: μ1 ≠ μ2 (group 1 mean is different from group 2 mean).
* Use case: use a two-tailed t test when you want to determine if there is any difference, regardless of the direction
Key differences:
* One tailed t-test is directional, while two-tailed is non directionalA significant result in a one tailed test confirms the hypothesis in one direction, whereas a significant result in a two tailed test indicates a difference, but does not specify direction
Degrees of Freedom:
* Based on the size of the sample
* n-1 for one group, or N-2 for 2 groups
* N for 2 groups, n for 1 group
Degrees of freedom = the number of independent values or quantities that can vary in an analysis without breaking any constrains.
Single sample: When you have a single sample of size n, the degrees of freedom is typically n-1. This is because one value can be derived from the others meaning if you know n-1 values, the last value is determined based on the mean
* EX: if you have 5 measurements and their mean is 6, the sum is 5x6 = 30. If you know 4 of the measurements, you can find the 5th measurement by substracting the sum of the known values from 30.
Two groups: When comparing two groups (like in a t-test), the degrees of freedom is calculated as N-2, where N is the total number of observations in both groups. This accounts for estimating two means (one for each group).
* EX: if you have two gorups, one with 3 subjects and the other with 4, N is 7, or the defrees of freedom would be 7 - 2 = 5
This is how we find the critical value
a1 = 1 tailed t test
a2 = 2 tailed t test
We go over to what our alpha is set to at the top (the EX below is 2 tailed t-test (a2) w/ a confidence of 0.5)
We have 18 degrees of freedom in this example (df = 18 on the left)
Go over and down
Since we had two groups of 10 our we use N - 2
* 20-2 = 18
* df = 18
N = 10+10 =20
Critical value = 2.101
if t > 2,101 = significant difference
So if t is greater than the critical value there is a significant difference between groups
From the example prior our crtical value = 2.101
* You can see the critical value is the dotted blue lines (denoting +/-)
* if T > critical value than its a significant difference
* Meaning that if T falls outside the cirtical value there is a significant difference in group means
Notice the below is 2 tailed (meaning it can fall on either side)
we found that T = +/-2.718 falling outside those critical value bracets meaning that the group means are statsitically different
* a = 0.5, meaning that were 95% sure that the two numbers actaully have statistically significantly different means.
* Its two tailed which is why it says 0.025 instead of 0.05
The significance (2 tailed) is basically saying that were there is a 0.014 chance were wrong (and since thats less than 0.05 were good with that)
* Were confident that theres a significant difference between the two group means
This is for a t test
This kind of chart will be on test
Levenes Test for equality of variances
* This is testing to make sure the variances are equal between the two groups tested in a t-test
* It will give you a p value on if the two groups are significantly different
* We want this p value of significance to be high meaning theres no difference between the variances of the two groups
* If the p value was low, that would be that there is a significant difference between the variances of the two groups
* note: alpha is typically 0.05 in these levene tests (if its greater than that that means theres no significant difference in the variance between the two groups)
* In the box below we get .419 = not significant = the two groups do not have significantly different variances, meaning we can assume equal variances
* So now that we know that, we can stick with the “equal variances are assumed” line
* So probs know the first boxes are for variances
we then how our t value
then our df of freedom (calculaed N - 2 = 20-2 = 18)
Then we have our p value comparing the two group means. This is 0.014, which is less than alpha (0.05), meaning that there is a signfiicant difference between the two groups.
So the above information means that there isnt a significant difference of the variability of the data sets, however, there is a significant difference in the means
Then the 95% confidence inteval is saying that were 95% confident that the difference between the groups is between 1.058 and 8.26
* Note: since this doesnt cross 0 it makes sense that we found a significant difference between group means (it was less than 0.05 so it was a significant difference in means)
* “were 95% sure that the new grip strength group gained at least 1 pound of grip strength and at most 8 pounds of grip strength”
* She is specifically going to ask on a test how to interpret the 95% confidence interval in words, like I just did above
* “we are 95% confident that the average distance between the new and standard splint groups was between 1.06 and 8.26 pounds”
* Interpret that we can only get better if we use the new splint - however, this is purely EBP, we need to keep in account finical concerns, clinical experience, evidence etc… theres multiple things to consider except just EBP rout
* “Evidence says its good, however, may not benefit client as much as a slightly cheaper one thats not quite as good would because it allievates finaical concern
Remember if variances were unequal we would use the second box
We can now reject the null hypothesis that there was no difference between groups, and use the alternative that there is a difference in group means (2-tailed so that difference can be either way)
* We conclude that there is a signfiicant difference between group means
* 95% confidence interval did not cross 0 (which is why it was significant)
* 5% risk of type 1 error accepted
Unequal variance example
I think you can look at that standard deviation from the mean group to determine which group is more variable as well (but im not positive)
So under Levene’s Test for equality of variances we find p = 0.038 with a being 0.05, meaning theres a statistically different variance between the two groups. (so the significance under levenes test is the one we look at to diliniate between the variances)
* meaning that one of the two data sets is significantly more variable than the other one
SO WE MUST LOOK AT THE SECOND ROW WHICH IS EQUAL VARIANCES CANNOT BE ASSUMED
* So theres a signfiicantly difference between the variability of the two data sets
So looking at the second row we see that our significance for the p value of the t-test (which is looking to see if there is a signfiicant difference between the means of the two groups) is .002 which is considered statistically significant
* t-test is saying that there is a signfiiant difference in group means, however, there is also a signficiant difference in variability between the two groups.
Looking under the 95% confidence interval, were 95% confident that the difference between the two groups is between 2.085 and 8.2149
* Once again looking at the second row to derieve these because we cannot assume equal variances
* Notice, because taht 95% confidence interval does cross 0 its most likely statistically different
Also, df is found using N-2 (because two groups are being compared).
* N = 15+10 = 25
* 25-2 = 23
Now because were utilizing the equal variances not assumed group that df (23) is multiples by some vaule to deirve (in this example it yielded 20.625), however, we don’t actually have to do this calculation.
Going to be a test question on what line to read
again, this is what unequal variances would look at (the standard deviation from the middle is much higher in the top one than the bottom)
wider = more vairability
narrower = less variability
Paired Samples T-Test example
Comparing 30 participants gait speed uphill vs flat
* Notice were comparing two measurements in the same person (paired)
* With a t-test i can only compare two things (cant add in a third)
Could also be a treatment
* take running 100m gait speed (pre-intervention)
* Then go through some LE strengthening intervention
* Then take running 100m gait speed (post intervention)
* And check and see if theres a significant difference between the two measurements
* so theres a pair of data for each participant and comparing two groups (t-test)
NOTE: an independent groups would mean two groups entirely independent from each other. Paired is the same group of people that have a pair of data.
NOTE: for paired sample t-test instead of looking at the difference in averages you’re looking at the different scores
* Looking at the measure of the differences between the scores in each pair
Assumes:
* Samples are truely random from a normally distributed population
* Data are of the interval/ratio level (meaning continuous)
Paired T-Test Example:
* Each participant has a pair of scores
* Were comparing two things (t-test)
This is looking at a group of 8 participants w/ LBP
* We want to know what this pillow does to their pelvic tilt angle
We Found that on average the pelvic angle w/ the pillow is 102.375 and without the pillow it was 99
D bar = average difference in pelvis angle of all the groups = 3.375 (this is essentially out x bar from prior)
* basically saying how different on average were the two measurements = D bar
* Its not the same as X bar because X bar is just the group mean, this is comparing individual differences in 1 person pre and post intervention and then deriving the the difference then taking the mean. (very similar though)
Because the pillow can only increase lumbar lordosis it means the pelvis can only go one way, meaning we can utilize a 1 way t-test
Null hypothesis = no change
Alterantive hypothesis H = u1 > u2
* note its only a one way t-test, meaning we utilize > or < symbole instead of a not equal to symbol
To find t in a pared t-test we do d bar over S dbar
S dbar = average variability
D bar = average difference (so finding the difference in scores pre and post intervention and taking the mean of them) - its this instead of x bar because its a pared test (meaning the same person is getting a pair of data)
We find that T = 1.532
Shes not going to have us calculate that but we do need to know the ratio for a paired t-test is to calculate t is D bar over S Dbar
* Look below to see it
KNOW: If there is a larger D bar (difference between group 1 and group 2) than T will be larger (because its on the top of the equation)
* This makes sense
KNOW: If variability of the data set increases T will decrease (because its on the bottom)
* This makes sense
if you’re looking at a T value thats not significant (is not greater than the critical value) for a pared t-test you can understand that either the D was small (not much of a difference after intervention) or there was a lot of variability between groups which made it weaker.
Remember if T < than critical value t is not significant and doesnt really matter
This is pulled from the example above
Because its a paired t-test meaning its 1 group of people w/ 2 data sets they can run a correlation between pillow and no pillow, meaning will the pillow always increase or decrease the pelvic angle. They found a moderate correlation of .708 = moderate
* so the more they use the pillow the more increase in pelvic angle they have
* And its just significant (0.049 < 0.05 = significant)
Mean = pillow - no pillow pelvic angle average difference mean - 3.375
95% confidence interval
* upper = 8.585
* lower = -1.84
* Notice this crosses 0 meaning that its likely not significant - were 95% confident that it could decrease or increase the pelvic angle (thats not really useful)
Significance = 0.169 = not singificant (makese since from above data)
* It says 2 tailed, thats the significance if we looked both ways (we only looked 1 way so she divided it by 2 or something? doesnt matter not going to test on this)
Df = 7
* We use the formula n-1 to find this because even though we have 2 measurements taken they’re both in 1 group so thats why we use -1
* N = 8
* 8 - 1 = 7
So this is looking at a two tailed T test where a = 0.05
* If were looking at a 1 tailed t-test were only looking at 1 side a = 0.25
* To keep a 0.05 we just divide p by 2 instead
So for a 1 tailed if we get p = 0.169 we need to divide this by 2 (which will actaully make it a bigger #) and if thats less than 0.05 its considered significant
However, this probs wont be on test
Our critical value is 2.1 so we need t to fall outside this critical value to be considered signficant.
The t we calculateed from the example above is 1.5
so that .169 significance is considering both sides (for a 2 tailed t test)
we divide that .169 by 2 because were not looking at the left half
Im honestly not sure about any of this shit
From the example prior we ran a 1 tailed t test (because the pillow can only increase lumbar lordosis) in the positive direction
Notice our 95% confidence interval crosses that 0, so its likely not signficant
df = 7
We did a 1-tailed t test (a1) at 0.5
Critical value = 1.895 (from matching the two #’s above on the chart)
her picture is fucked up but if t falls beneath the critical value meaning its not a significant difference in means between the two groups
Except the null hypothesis of no change
Does this have a large of small effect size?
Small
Which one has a larger effect size
B - the difference between these groups is larger
* how significant is that difference between groups
The effect size is basically saying how big the difference between the averages is after the intervention is applied.
For example, below C is bigger than B, meaning the effect size in C was larger. In other words, the independent variable affected the dependent variable more in C than it D in B
* The size of the effeect size is expressed in standard deviations (B is 1 standard deviation away from the mean [smaller effect size] where as C is 2 standard deviations away from the mean [slightly larger effect size])
* AKA standardized mean difference (SMD) - how many standard deviatinons you are away from the original mean
* These calculations are based on the t-test performed - meaning thats how we found the average distance in means away from eachother (the variance within each group is going to be roughly equal because this is a t-test, were just looking at how far away the mean moved following the intervention to derieve the effect size)
So after running the t-test it shows the group mean of the new splint group was ~pounds more than the standard
* So the new splint increased their mean by 10.1100
d = (xbar1 - xbar2)/s
s = the common standard deviations of the means
* So they just took the average of the standard deviations (3.72+3.94)/2 = 3.83
* X bar = the mean
* And we find that d = 1.22, meaning that these two groups differ by 1.22 standard deviations (which is the effect size of using the new splint over the standard) - the new splint yields 1.22 standard deviations more grip strength
* this is a large effect size (large = 0.8-1)
* I guess the effect size of large is more like 0.8+
* the effect size being large means that most people that use this new splint are going to have more grip strength gains than those that use the standard
* the effect size tells us what happens to most individuals in that situation
* If its a small effect size the difference between group means may be significant, but it may not happen for everybody - a large effect size would say that this difference is proably going to happen for everyone
She wants us to be able to read the results as well - mostly in the abstract
New splint group increase in pinch strength from baseline = 10.11 pounds +/- 3.62 pounds (meaning it could deviate either direction buy that much
Standard group increase in punch strength from baseline = 5.45 pounds +/- 3.95 pounds
The mean difference between groups at the end of the study was 4.66 pounds, meaning the group average is each group will different by 4.66 pounds (New splint gained 4.66 pounds more grip strength than standrd)
The average distance between the two studies was 4.66 pounds (meaning that the delta between their averages was 4.66 pounds) and were 95% confident that the true number for the average distance between the two groups is between 1.06 and 8.62
* likely a significant difference because it doesnt include 0
An independent, two tailed t-test showed that the group using the new splint had a significantly larger change in strength than the group using the standard splint (t = (18) = 2.718, p = 0.014, d = 1.2)
* p = 0.014 which is < a = 0.05, meaning that our proabibility of type 1 error (identifying a difference when there is none) is very low so we accept it
* d = 1.2, meaning that there is a large effect size (meaning the independent variable has a large effect on the dependent variable and most people will see a difference)
* t = 2.718, however, we can’t really do much w/ this because we don’t have a critical value to compare it two
* Independent two tailed test, meaning that two seperate groups were used (not paired), comparing two group means independent of each other. Two tailed, meaning it could go both ways, one group could be higher, one group could be lower, we arent sure they will necessarily increase in pinch strength
Effect size: Pared t-test
Effect size is still depicted w/ cohren’s d
* d bar = mean differences in scores - meaning each subject gets two scores and you’re looking at the average difference in all the subjects scores from before and after the intervention
* Sd = the standard deviation of the difference scores
She’s not going to have us calculate this, she just wants us to know that the way you calculate effect size is different between a paired t-test and an unpared t-test
* how you calcualte that effect size depends on the type of t-test you’re doing
Wehn doing a post-hoc (determining power) analysis for a paired t-test (note: the one above was an unpaired t-test)
The only difference is when you find the d (effect size) we divide it by radical 2
* with independent t-test we just used d
NOTE: when shes asking about d in a paired t-test we still interepet it the same way, (small, medium, large) w/o dividing by radical 2. However, its ONLY WHEN CONDUCTING THE POWER ANALYSIS OR POST HOC TEST that we divide by radical 2, all the other times we see d we don’t divide by radical 2
All the other variables are the same
EX: What 3 variables are used to estimate power size on a paired t-test:
* a
* d/radical2
* n
* probs also note if its 1 tailed vs 2 tailed
This is specifically for a paired t-test
only difference is we use d/radical 2 instead of d
The mean difference in pelvic tilt angle between using and not using a lumbar pillow was 3.38 degrees +/- 6.23 (95% CI = -1.84 to 8.58)
* saying on average the difference between the two groups was 3.38 degrees (meaning the group means were that far seperated)
* however, they’re 95% that true group mean difference is between -1.84 to 8.58
* Notice this crosses 0 so its proably not significant
* “On average the pillow tilted the pelvis 3.38 more degrees”
A paired, one-tailed t-test showed that there was no significant difference between the two test conditions (t (7) = 1.532, p = .169, d = .77)
* one tailed t test = hypothesis is only going on direction (pillow can only push pelvis on direction)
* Paired, meaning its the same group of people for both conditions
* t = 1.532, doesnt do us much good because theres no critical value to compare it to
* p = 0.169 = > 0.05 = not significant
* d = .77 = moderate effect size (a decent number of subjects will see a change between using the pillow and not using it)
A post hoc analysis (after the study) showed that the test achieved 40% power
* they used a, d/radical2, n to derieve this
* Meaning that were 40% sure that we correctly identified a change (type 2 error)
So even though we found a moderate effect size the a and power both suck so its not really great
