# Interpretation of interaction effect in factorial experiment

Hello

Please help me understand how to interpret the results of full-factorial experiment with 4 factors when 1 of 2 main effects is not significant, but their interaction is significant.

My dataset:

`````` A B C D Trials Succeses
1  0 0 0 0   1852       11
2  0 0 0 1   1878        3
3  0 0 1 0   1869        9
4  0 0 1 1   1881       14
5  0 1 0 0   1926        4
6  0 1 0 1   1920        6
7  0 1 1 0   1891        4
8  0 1 1 1   1841        5
9  1 0 0 0   1921        9
10 1 0 0 1   1827        2
11 1 0 1 0   1837       13
12 1 0 1 1   1908       11
13 1 1 0 0   1827        8
14 1 1 0 1   1860        5
15 1 1 1 0   1854       10
16 1 1 1 1   1922       10
``````

My final model is (Succeses, Trials) ~ (C + D + C * D)

``````glm(formula = cbind(Succeses, Trials) ~ (C + D + C * D), family = binomial(link = logit),
data = df)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-1.9108  -0.6633   0.1795   0.5882   1.2902

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -5.54543    0.09587 -57.843  < 2e-16 ***
C1          -0.25879    0.09587  -2.699  0.00695 **
D1           0.14895    0.09587   1.554  0.12028
C1:D1        0.19490    0.09587   2.033  0.04206 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 27.866  on 15  degrees of freedom
Residual deviance: 16.007  on 12  degrees of freedom
AIC: 84.463

Number of Fisher Scoring iterations: 4
``````

Model with all interactions:

``````glm(formula = cbind(Succeses, Trials) ~ (A + B + C + D)^5, family = binomial(link = logit),
data = df)

Deviance Residuals:
[1]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -5.620799   0.104319 -53.881   <2e-16 ***
A1          -0.105993   0.104319  -1.016   0.3096
B1           0.109620   0.104319   1.051   0.2933
C1          -0.259357   0.104319  -2.486   0.0129 *
D1           0.150130   0.104319   1.439   0.1501
A1:B1        0.166693   0.104319   1.598   0.1101
A1:C1        0.108473   0.104319   1.040   0.2984
A1:D1       -0.122721   0.104319  -1.176   0.2394
B1:C1       -0.165999   0.104319  -1.591   0.1116
B1:D1        0.166955   0.104319   1.600   0.1095
C1:D1        0.205677   0.104319   1.972   0.0487 *
A1:B1:C1    -0.015377   0.104319  -0.147   0.8828
A1:B1:D1     0.025085   0.104319   0.240   0.8100
A1:C1:D1    -0.006925   0.104319  -0.066   0.9471
B1:C1:D1     0.169022   0.104319   1.620   0.1052
A1:B1:C1:D1  0.069391   0.104319   0.665   0.5059
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 2.7866e+01  on 15  degrees of freedom
Residual deviance: 1.0081e-13  on  0  degrees of freedom
AIC: 92.456

Number of Fisher Scoring iterations: 4
``````

Model with 2-way interactions

``````glm(formula = cbind(Succeses, Trials) ~ (A + B + C + D)^2, family = binomial(link = logit),
data = df)

Deviance Residuals:
1        2        3        4        5        6        7        8        9       10       11       12       13       14       15       16
0.4816  -0.8870  -0.3733   0.4042  -0.5775   0.7016   0.4976  -0.5010   0.2084  -0.2152  -0.2617   0.2079  -0.2944   0.2655   0.4013  -0.2842

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -5.60250    0.10146 -55.219   <2e-16 ***
A1          -0.09102    0.09661  -0.942   0.3461
B1           0.14402    0.09658   1.491   0.1359
C1          -0.23302    0.09782  -2.382   0.0172 *
D1           0.12723    0.09791   1.299   0.1938
A1:B1        0.18278    0.09613   1.901   0.0572 .
A1:C1        0.12331    0.09800   1.258   0.2083
A1:D1       -0.13583    0.09594  -1.416   0.1569
B1:C1       -0.14472    0.09857  -1.468   0.1420
B1:D1        0.13080    0.09692   1.350   0.1771
C1:D1        0.22362    0.09918   2.255   0.0242 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 27.8660  on 15  degrees of freedom
Residual deviance:  3.2462  on  5  degrees of freedom
AIC: 85.702

Number of Fisher Scoring iterations: 4
``````

I would be extremely grateful if you shed light on my 3 questions

1. I know that if there is a significant interaction effect then we should include it in a model even though one of the main effects may not be significant. As it is the case.
But how can we interpret the fact that when С at 0 level and D at 1 - we have a decrease in success rate by 50%. However, when С and D both at level 1 they increase success level by 20%. How to report this to stakeholders?

2. How confident can I be that factor С has a positive effect?
What confuses me is that when I look at a model with all interactions included then if factor C at 1 level it decreases the predicted success rate by -18%.
When I look at a model with 2-way interactions it increases success rate by 6%.

3. What are my next steps to make a clear conclusion?
Do I need to accept С factor as a most successful factor and run a follow-up experiment with factor D against control which will have factor C?

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