# How to calculate confidence interval for main and interaction effects? (2^3 factorial design with no replicates)

I have a three-factor (1, 2, 3), two-level (-, +) factorial design shown below (contrast coefficients for interaction effects and response parameters shown here as well). The overall goal is to determine which effects are significant and what their respective confidence intervals are. I've figured out how to determine which factors are significant but I can't figure out how to calculate the confidence intervals.

            |    Design     |     Interactions   |Response
Experiment  |  1    2    3  |  12   13   23   123|  Y
------------|---------------|--------------------|--------
1  |  -    -    -  |  +    +    +    -  |  12
2  |  +    -    -  |  -    -    +    +  |  9
3  |  -    +    -  |  -    +    -    +  |  17
4  |  +    +    -  |  +    -    -    -  |  18
5  |  -    -    +  |  +    -    -    +  |  16
6  |  +    -    +  |  -    +    -    -  |  8
7  |  -    +    +  |  -    -    +    -  |  14
8  |  +    +    +  |  +    +    +    +  |  18


I calculate the effects and model coefficients (for the main effects and all interaction effects) from the average response value at the - and + levels according to the above table in order to get:

Factor  |  Yave (-)  |  Yave (+)  |  Effect (Y+ - Y-)  |  Coefficient
--------|------------|------------|--------------------|-------------
1  |  14.75     |  13.25     |  -1.5              |  -0.75
2  |  11.25     |  16.75     |  5.5               |  2.75
3  |  14        |  14        |  0                 |  0
12  |  12        |  16        |  4                 |  2
13  |  14.25     |  13.75     |  -0.5              |  -0.25
23  |  14.75     |  13.25     |  -1.5              |  -0.75
123  |  13        |  15        |  2                 |  1


I find the sum of squares and the degrees of freedom for each factor by doing an ANOVA between the two levels:

          |            |Total from| Between  |  Within
|Observations|Grand Ave |  Levels  |  Levels
|  -     +   |  -    +  |  -    +  |  -    +
----------|------------|----------|----------|-------------
|  12     9  |  -2   -5 |0.75 -0.75|-2.75  -4.25
|  17    18  |  4    3  |0.75 -0.75|2.25   4.75
|  16     8  |  -6   2  |0.75 -0.75|1.25   -5.25
|  14    18  |  0    4  |0.75 -0.75|-0.75  4.75
----------|------------|----------|----------|-------------
Treat Ave |14.75  13.25|          |          |
Grand Ave |  14    14  |          |          |
Difference| 0.75  -0.75|          |          |
----------|------------|----------|----------|-------------
Sum Square|            |   110    |   4.5    |    105.5
Deg Freed |            | 8-1 = 7  | 2-1 = 1  |(4-1)+(4-1)=6


Due to the residual here being due to both error and the aggregate variability of all other factors, an F test was not performed here to determine if factor 1 was significant. Instead the between level sum of squares and degrees of freedom was kept for later, and the same procedure was carried out for all factors, which I've summarized in the following table:

Factor | Deg Freed | Sum Squares | Mean Squares
-------|-----------|-------------|--------------
1 |     1     |     4.5     |     4.5
2 |     1     |     60.5    |     60.5
3 |     1     |     0       |     0
12 |     1     |     32      |     32
13 |     1     |     0.5     |     0.5
23 |     1     |     4.5     |     4.5
123 |     1     |     8       |     8
-------|-----------|-------------|--------------
error |    N/A    |     N/A     |     N/A
total |     7     |     110     |


I assumed that some factors were insignificant and created an error estimate by pooling factors together until the factor with the smallest non-zero sum of squares was included which results in:

Factor | Deg Freed | Sum Squares | Mean Squares | F Stat
-------|-----------|-------------|--------------|--------
1 |     1     |     4.5     |     4.5      |  18
2 |     1     |     60.5    |     60.5     |  242
3 |  pooled   |   pooled    |    pooled    | pooled
12 |     1     |     32      |     32       |  128
13 |  pooled   |   pooled    |    pooled    | pooled
23 |     1     |     4.5     |     4.5      |  18
123 |     1     |     8       |     8        |  32
-------|-----------|-------------|--------------|--------
error |     2     |     0.5     |     0.25     |
total |     7     |     110     |              |


Each remaining factor has 1 degree of freedom and the error term has 2 degrees of freedom. At the 5% significance level F(0.95, 1, 2) = 18.51. Since factor 1 and 23 have F values less than the critical value, they are now considered insignificant as well and pooled into the error term. This process was repeated until I converged on the following table:

Factor | Deg Freed | Sum Squares | Mean Squares | F Stat
-------|-----------|-------------|--------------|--------
1 |  pooled   |   pooled    |    pooled    | pooled
2 |     1     |     60.5    |     60.5     | 17.29
3 |  pooled   |   pooled    |    pooled    | pooled
12 |     1     |     32      |     32       | 9.14
13 |  pooled   |   pooled    |    pooled    | pooled
23 |  pooled   |   pooled    |    pooled    | pooled
123 |  pooled   |   pooled    |    pooled    | pooled
-------|-----------|-------------|--------------|--------
error |     5     |    17.5     |     3.5      |
total |     7     |     110     |              |F(0.95, 1, 5) = 6.61


At this point I feel confident concluding that factor 2 (effect = 5.5) and factor 12 (effect = 4) are significant (my results match the example problem I've been following along with and they also match if I change the response values from other examples online). However, I'm pretty lost as to how to determine the confidence intervals for them. I've attached an image of the confidence interval calculation from the example problem I've been following along with (I haven't been able to find other example problems online)

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