Data were collected on TBUT, a measure of how long it takes...

Key Concepts

Intraclass Correlation Coefficient (ICC)

The ICC is a statistic that quantifies the proportion of the total variability in measurements that is due to differences between subjects as opposed to measurement error or within?subject variability. It is commonly used in reliability studies to assess how consistent or reproducible measurements are when repeated under similar conditions.

Variance Components

Variance components analysis involves partitioning the overall variability in measurement data into between-subject variability (which represents true differences among individuals) and within-subject variability (which represents error or natural fluctuations in repeated measures). The ICC is essentially the ratio of between-subject variance to the total variance, making this decomposition fundamental to its calculation.

Measurement Reproducibility

Measurement reproducibility refers to the consistency of a measurement when repeated under the same conditions. High reproducibility implies that most of the observed variance in measurements is due to actual differences between subjects rather than random error, which is what the ICC captures.

Repeated Measures Study Design

A repeated measures study design is one in which multiple measurements are taken from the same subject under identical or similar conditions. This design is crucial in reliability studies because it allows for the assessment of the consistency of the measurement across time or replicates, facilitating the calculation of metrics like the ICC.

Transcript

00:01 All right, this question says that the data in the following table represents the number of hours of relief provided by five different brains of headache tablets administered to 25 subjects experiencing fever at 38 degrees celsius of every sick or more.

00:23 All right.

00:25 Performing the analysis of virus.

00:29 Okay, it should be performed.

00:31 Perform the analysis of variance and test the hypothesis at the 0 .05 level of significance that the mean number of hours of relief provided by the tablets is the same for all five brains.

00:46 Discuss the results.

00:49 Okay, so we are using the anova there, right, we have five different brands there, a, b, c, d and e brains there.

00:56 So you can see that each brand or each group has five, right? data sets there so all in all they are 25 okay so now let's try to see how we go about it so our new sorry our hypothesis are that the null hypothesis is saying all the means are equal so mean one is going to mean two is up to mean five then and the alternative hypothesis says at least two of the means are not equal there and our significance level is 0 .05 there and our f figure that you calculate that they're using um 4 degrees of freedom and 20 decrease of freedom this is the denominator and this is the numerator there so by the way how do we get this one this is k minus 1 so k minus 1 is course to 4 right so and this one is n minus k so if k this is k minus one which is equal to four which is five minus one is got the four and then this one is 25 25 minus k right which is equal to 20 okay so our k is the value a per group there so that is the number of groups that we have it's five groups but 25 is the total number of total says that we have minus the number of groups that we have which is k there and then it gives us 20 degrees of freedom there all right so and with that we get our 2 .87 there on the table so this is the column and this is the raw right when you're looking at the table this four you look you go to the columns going downwards and this one going side with the rows okay right so when we do that and then i just put on a simple table after doing all the calculations so for the treatment figure the simple the sum of squares that i found is 78 .22 and then 59 .532 there for the error there then the decrease of freedom is 4 and 20 you can see that so for us to get the main squares there you have to divide for the treatment you divide 78 2 2 divided by 4 you get 19 60 5 5 there and then for the main square of the error you divide 20 59 532 divided by 20 we get 2 .97 and then to get the computed f there we divide right 19 divided 19 .605 divided by two comma 97 there and we get our 6 .59 there.

03:46 Okay, so i think the main question will be how exactly do we find these two figures because these two figures they then give us all of the figures there.

03:54 These ones we've already found them.

03:56 So how do we get the treatment, the sum of squares of the treatment and the sum of squares of the right? so i'll show you the calculation just below.

04:04 But this is generally leading us to the actual answers that we want so now our p value they've completed our p value can use the software you can use the tables so our p value is 0 .015 there right then the decision that we make is that we have to reject the now hypothesis there why are we reaching the now hypothesis you can see the v that we've calculated them right and the value that we found from the tables there the f value that we have there from the critical region there right the f from the critical region there is what this is two comma right this value here is smaller than the value that we have calculated right so the value that we've calculated is greater so it means we have to reject right so we therefore we are rejecting the null hypothesis there from the critical region that we are rejecting the null hypothesis in other ways we are saying the main number of hours of relief differ significantly they are different okay they're not the same so but okay let's go back i wanted us to find out these two values there how do we get 78 .5 and how do we get 59 .53 there which are the sum of squares of treatment and error there so first of all let the number the total number of the population in there the population number is 25 but for each group it's five per group so we have to add the total for the first one the total for the second one we get a blueprint different from the right to the first total and then for a for b then the total for c the total for c if we add all these totals i found 137 comma three there and i say this is my capital t there so i will have to use my capital t somewhere there okay then uh i have this sum of t squared so right the sum of t i so when t is one we have to do t2 up to t 25 there so if you add all those squares you get right this is what i have found there okay all right so now this is the sum of the t1 squared t2 squared up to t5 squared there so this is what we we get there now right so but for us to get um the sse, the sse is, would be the figure for treatment there, the treatment, some of skills for treatment.

06:45 And then the s s, right, t .r there.

06:52 Okay, let's see.

06:54 Right.

06:55 The sstr would be our figure for the treatment, right? this is some treatment and this is for the error.

07:02 You can see if this is some of error there.

07:05 And then there's of treatment.

07:06 So the sum of right.

07:09 So to get s s ewe, first of all find sst.

07:13 So but sst there, it is the sum of this t squared.

07:16 We found the two squared, which is 8 .889...