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A difference of means with both halves uncertain, standard errors combined in quadrature, and the paired design that turns the whole problem back into a one-sample one.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will decide from a design whether two groups are paired or independent, combine two standard errors into the standard error of their difference, count the degrees of freedom each design has, and analyse a paired study by forming its differences and running the one-sample procedure on them.
Every test so far compared one estimate with a number somebody wrote down. Most real questions compare two groups, so the quantity of interest is a difference and both halves of it are uncertain. The only new arithmetic is that independent variances add.
A design is paired when each observation in one group has a natural partner in the other, decided by the design: the same subject twice, twins, two measurements of one item. Otherwise the samples are independent. Combining two standard errors by squaring, adding and taking the square root is called combining in quadrature. A pooled estimate of the spread assumes the two groups have the same variance.
Independent samples. The estimate is $\bar x_1 - \bar x_2$, and because the two means come from different people their errors are independent, so the variances add:
$$\text{SE}(\bar x_1 - \bar x_2) = \sqrt{\text{SE}_1^{2} + \text{SE}_2^{2}}.$$
Adding the standard errors instead overstates the uncertainty by up to about forty per cent. The statistic is the difference over that standard error, and the degrees of freedom are $n_1 + n_2 - 2$ when the two spreads are pooled, or a fractional Welch value when they are not.
Paired samples. Here the two readings on a subject are not independent — a large subject is large in both — so the formula above does not apply. The repair is not a harder formula but a simpler one: replace each pair by its difference, and analyse the resulting single sample exactly as in the last four lessons, testing its mean against zero on $n - 1$ degrees of freedom.
Pairing is usually a large gain. Everything that differs between subjects cancels in the difference, so the standard error can fall by a factor of several. Ignoring pairing that exists throws that away; claiming pairing that does not exist invents a correspondence the data have not got, and produces a standard error far too small. The design decides which, and the numbers never can.
Another way: story
To find out whether a diet works, you can weigh a hundred people on the diet and a hundred others not on it, or you can weigh the same hundred people before and after. The first comparison is swamped by the fact that people differ enormously in weight. The second never sees that variation at all, because it cancels in every subtraction — which is why the second study can be half the size and twice as informative.
Another way: steps
Two standard errors, and the standard error of their difference.
| First | Second | Sum (wrong) | In quadrature (right) |
|---|---|---|---|
| 3 | 4 | 7 | 5 |
| 5 | 5 | 10 | 7.07 |
| 1 | 5 | 6 | 5.10 |
| 5 | 12 | 17 | 13 |
The third row is the one to remember. When one group is far noisier than the other, the answer is barely larger than the noisier one alone — so improving the quieter group buys almost nothing, and every extra observation belongs in the noisy group. A design rule falls straight out of the arithmetic.
Adding the two standard errors. Variances add, not standard errors. The sum is always too large.
Subtracting the two standard errors. Subtracting the means does not subtract the uncertainties. Both estimates are uncertain and both uncertainties survive.
Analysing a paired design as two independent samples. It is valid and it is wasteful: the between-subject variation that pairing removes is put back in, and the test loses most of its power.
Analysing independent samples as if paired. This is not merely wasteful, it is wrong. It invents a correspondence and reports a standard error that is far too small.
Deciding from the numbers. Whether a design pairs is a fact about how the data were collected. Equal group sizes are not evidence of it.
Group A: $\bar x = 52$, standard error $1.8$. Group B: $\bar x = 47$, standard error $2.4$.
Different people in each.
Difference $5$; standard error $\sqrt{1.8^{2} + 2.4^{2}} = \sqrt{3.24 + 5.76} = 3$.
Quadrature, not addition.
Statistic $5/3 = 1.67$, which a two-sided test at $5\%$ does not reject.
Difference over its standard error.
Twenty subjects measured twice; the differences have mean $5$ and standard deviation $4$.
One sample of twenty differences.
Standard error $4/\sqrt{20} \approx 0.894$, statistic $5/0.894 \approx 5.6$ on $19$ degrees of freedom.
The one-sample procedure.
Comfortably significant, from a study a fifth the size — because the subjects' own variation never entered.
This is what pairing buys.
Square and add: $64 + 36$.
Variances add.
That is $100$.
So the standard error of the difference is $10$, not $14$.
Consider the same twenty patients measured before and after a treatment. Is this a paired design?
Two independent groups give sample means whose standard errors are $15$ and $20$. What is the standard error of the difference between them?
Answer:
Group one has mean $67$ with standard error $27$; group two has mean $56$ with standard error $36$, and the groups are independent. Give the estimated difference, and its standard error.
Difference: g. Standard error: s.
Three comparisons, each between two independent groups. For each pair of standard errors, give the standard error of the difference.
| Standard error of the difference | |
|---|---|
| Standard errors $27$ and $36$ | |
| Standard errors $54$ and $72$ | |
| Standard errors $45$ and $108$ |
$20$ subjects are each measured before and after a treatment. Put the steps of analysing the study into the order they are carried out.
Number the steps in order (write the number in the box):
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Four comparisons of a mean around $58$, each analysed with a t procedure. Match each design to its degrees of freedom.
| $19$ degrees of freedom | $14$ degrees of freedom | $38$ degrees of freedom | $28$ degrees of freedom | |
|---|---|---|---|---|
| $20$ subjects measured before and after | ||||
| $15$ matched pairs, one of each pair in each group | ||||
| Two independent groups of $20$, pooled | ||||
| Independent groups of $12$ and $18$, pooled |
You can tell a paired design from two independent samples, combine two standard errors correctly, and turn a paired study into a one-sample analysis. Say in your own words why pairing is decided by the design and never by the numbers.
9. Your turn: independent groups with standard errors $8$ and $6$, step 3