1W ANOVA. Each combination group-antibiotic type is a treatment. First you perform 1W-ANOVA and then (if exist significance) a Tukey test in order to find significance difference between treatments. It is extremely important that the whole of your treatments and replicates were randomly executed and the experimental subjects were randomly assigned to treatments and replicates.

Now your treatments.

I guess that 1 represent two treatments: A without antibiotic (Treat 1), B without antibiotic (Treat 2). I mean without antibiotic as no-treatment in order to avoid confusion with statistical treatments for 1W-ANOVA.

I guess that 2 must be group A with antibiotic 1 (GA+Ant1: Treat 3). Because (Group A no treatment) is Treat 1.

The following treatments must be: (GA+Ant2: Treat 4), (GA+Ant3: Treat 5), (GB+Ant1: Treat 6), (GB+Ant2: Treat 7) and (GB+Ant3: Treat 8).

In this case you have 8 treatments in which you must assigned the mice A randomly in Treatments 1, 3, 4 and 5 (and replicates), and the mice B randomly in Treatments 2, 6, 7 and 8 (and replicates).

I guess that the second group represents comparison test: GA+Ant1 vs GB+Ant1 and so on.

You must perform 1W-ANOVA for your 8 treatments and Tukey comparison for the specific differences between treatments, including the comparison in your second group.

I think there are some interesting research questions that lead to these groups and treatments, like:

- how do the mice (of either group A or B) react on ab1?,

- how do they react on ab2?,

- how do they react when ab1 and ab2 are given together (do the effects only add up or is there a synergistic effect)?

and

- how different are all these reactions between the mouse groups?

These questions can be adressed using a model that predicts the response (you did not specify what the response is! - it can be recovery, survival, some biochemical, pharmacolocical or immunological parameter, or whatsoever) by a function of the predictors that are given as:

a) mouse group (can be A or B)

b) ab1 was applied (can be yes or no)

c) ab2 was applied (can be yes or no)

These are all factors, so this will be a three-factorial model. The questions above can be answered when the model includes all possible interactions (there are 3 two-way interactions and 1 three-way interaction). All these "how much" questions are adressed by coefficients in that model that are estimated based on the available data. These estimates are associated with uncertainly measures (standard errors) so that you can test specific hypotheses about the values of these coefficients.

To estimate the coefficients you need some probability model about the data (because the coefs will be chosen to maximize the probability [likelihood] of the observed data under that probability model). What an appropriate propability model may be depends on the response. If the response is something that can take two values (like survival or recovery [yes/no]) then a binomial model is in charge. If the response is a count value (numbers of some histological features, numbers of immune cells within certain structures, etc) a Poisson model or a negative binomial model will be appropriate, if the response is a proportion (e.g. the proportion of CD8+ cells among all lymphocytes) then a beta-model is the best choice, if it is a concentration then a gamma-model or a log-normal model will be appropriate and so on.

The models are known as "linear models". The standard or general linear model uses the normal probability model to estimate the coefs. The generalized linear models allow to specify a probability model from the exponential family, like binomial, Poisson or gamma. Linear models with other probability parts (eg. beta or negative binomial) ore sometimes listed separately in some software, but only because the fitting algorithms are different, but these are also "linear models".

I totally agree with you Wilhelm and Miguel, that we have more than two factors and that she is interesting in over-all difference among the groups. First she has to test for normality of the data. Second, how many mice in each group (n = ??). If you have a low number of animals and not normally distributed other than ANOVA test is recommended as Kruskall Wallis test.. Therefore, a test for an over all difference among the groups should be tested, than a test for differences between individual groups is carried out.

@Fathi

note that you only partially agree with Miguel but not at all to me. At least do I not at all agree with your answer.

Resuming the experimental design that I´ve suggested,

Treatment 1: Group A without antibody

Treatment 2: Group B without antibody

Treatment 3: Group A + antibody 1

Treatment 4: Group A + antibody 2

Treatment 5: Group A + antibody 3 (antibody 1 + antibody 2 in agreement with your nomenclature).

Treatment 6: Group B + antibody 1

Treatment 7: Group B + antibody 2

Treatment 8: Group B + antibody 3.

Then, the 3 question planted by Jochen can be answer with 1W-ANOVA and the adequate pair comparisons. Assuming that the 1W-ANOVA of the 8 treatments show that at least one treatment is different to others (1W-ANOVA significant). Jochen´s questions will be answered as,

 How do the mice (of either group A or B) react on ab1?.

By Tukey comparison of (Treatment 1 media - Treatment 3 media) and (Treatment 2 media - Treatment 6 media).

 How do they react on ab2?.

By Tukey comparison of (Treatment 1 media - Treatment 4 media) and (Treatment 2 media - Treatment 7 media).

 How do they react when ab1 and ab2 are given together (do the effects only add up or is there a synergistic effect)?

By Tukey comparison of (Treatment 1 media - Treatment 5 media) and (Treatment 2 media - Treatment 8 media).

In whole cases I guess that your response will be a variable that can be represented in real numbers (a measurable variable instead a countable variable). However if your response variable is countable Prof. Fathi has reason: You must performed Kruskall Wallis test.

Additionally you can use a linear model like the following,

Y=b0+b1x1+b2x2+b12x1x2+b3x3+b13x1x3+b23x2x3+b123x1x2x3    Eq. (1)

Where Y is your response or an adequate transformation (log or logit) or your response; x1 is antibody 1 concentration (0 when no treatment); x2 is antibody 2 concentration; and, x3 is a Boolean variable that takes the values: x3=1 (Group A) and x3=0 (Group B). You must fit by linear regression Eq. (1) to your experimental results and you must built null hypothesis (Ho) in order to evaluate the effects. As example,  

How do the mice (of either group A or B) react on ab1?.

Ho (Antibody 1 has not effect in either Group A or B): b1=0, b13=0, b12=0 and b123=0 (simultaneously). If you reject Ho you have statistic evidence that Antibody 1 has effect in either Group A or B.

With respect to the same antibody,

Ho (Antibody 1 has not effect in Group A): b1+b13=0 and b12+b123=0 (simultaneously). If you reject Ho you have statistic evidence that Antibody 1 has effect in Group A. b1+b13 and b12+b123 are the Eq. (1) coefficients that relate the antibody concentration with your response.  

Ho (Antibody 1 has not effect in Group B): b1=0 and b12=0 (simultaneously). If you reject Ho you have statistic evidence that Antibody 1 has effect in Group B. b1 and b12 are the Eq. (1) coefficients that relate the antibody concentration with your response.  

Similar for antibody 2 but with b2, b23, b12 and b123 parameters.

How do they react when ab1 and ab2 are given together (do the effects only add up or is there a synergistic effect)?

Ho (There is not synergistic effect of antibodies): b12=0 and b123=0 (simultaneously). If you reject Ho, you have statistic evidence that there is synergistic effect of antibodies.

The simultaneous test for parameters (b1, b12,…) must be done with statistic for general linear models. You can find in,

Neter, J. and Wasserman, W. (1978). "Applied Linear Statistical Models" Ed. Richard D. Irwin, Inc. Homewood, Il.

I you want, I have a MatLab code to perform any Ho (in matrix form) for any linear model.

Yes, it is ok. The 8 treatments are the 8 groups that you refer. In the case of linear model (Eq. 1) x3 is a Boolean variable that takes into account the difference between A (x3=1) and B (x3=0). Each treatment has 5 replicates. You can use Eq. (1) to perform the statistical analysis of your results.

So I can not use the t test to compare each 2 groups, right? The better choice will be do perform a 2-way anova for different groups same treatment? and also for same groups different treatments?

I shouldn't use one way anova for GROUP A treatment 1 x GROUP A treatment 2 because eventhough they are the same group, I have two conditions: groups and treatments, right? Or should I use a one way for this one? 

No, a t-student test is not applicable because you have many different treatments. Rigorously, you have 3 factors with two levels each: groups (A and B), antibody 1 (0, and certain concentration), antibody 2 (0, certain concentration). Then you need a 2x2x2 factorial design (the design with 8 treatments detailed in my second response) randomly performed (the 5 animals used in each treatment must be randomly assigned). Then you must use a 3Way ANOVA but alternatively you can use too 1W-ANOVA and you can use the linear model (Eq. 1). The three analyses are equivalent. 3W-ANOVA calculates:

the square sum of variation due to groups (SSG),

 square sum of variation due to antibody 1 (SSA1),

 square sum of variation due to antibody 2 (SSA2),

 square sum of variation due to interaction group-antibody 1 (SSGA1),

 square sum of variation due to interaction group-antibody 2 (SSGA2),

 square sum of variation due to interaction antibody 1-antibody 2 (SSA1A2);

 square sum of variation due to interaction group-antibody 1-antibody 2(SSGA1A2).

In total 7 sources of variation, and like each effect has two levels each source of variation take 1 degree of freedom:7 degrees of freedom for square sum of source of variation. 1 degree of freedom for the great mean. Then the square sum of residuals (SSR) will have (N-8) degrees of freedom (N=5x8=40).

1W ANOVA calculates the square sum of treAtments (SSA) as the unique source of variation, but like there is 8 treatments SSA has 7 degrees of freedom (the same 7 that 3W-ANOVA), and 1 for great media. SSR will have (N-8) degrees of freedom (the same that 3W-ANOVA).

Actually if you have software you can confirm that (I can demonstrate it with mathematics but the space in not enough):

SSA=SSG+SSA1+SSA2+SSGA1+SSGA2+SSA1A2+SSGA1A2

(SSA square sum of treAments in 1W-ANOVA).

That is, 1W-ANOVA is fully equivalent to 3W-ANOVA. The difference is that 3W-ANOVA can indicate to you prior to multiple comparison test if exit significance of the any of source of variation. 1W-ANOVA group the whole source of variation in only SSA, but the multiple comparisons can indicate the differences between different combinations of the three factors.

I prefer 1W-ANOVA because I don’t like comercial software, I prefer use my own programs (which I programed 30 years ago), and like 1W-ANOVA is equivalent to nW-ANOVA is more easy to me. But if you have commercial software, perform 3W-ANOVA.  

Finally, if you count the number of parameter of Eq. (1) in my second answer, you will count 8. The same 8 of the source of variation and great media for 3W-ANOVA; or the same 8 for the SSA and the great media of the 1W-ANOVA.  

I am using the GraphPad Prism so I will just use the 1W ANOVA. 

You should not let a software dictate the analysis. You should chose the analysis that is (most) appropriate to answer your scientific question.

Bruna

In my opinion this is the better election. Remember the most important are the multiple comparison tests. Through multiple comparison tests (Tukey is the preferred) you can deduce the whole of the effects that you need between factors and interactions. 

SPSS cross tab analysis may fit for this data

Hi, Bruna

If the two different mice groups are different two strains it should be a factor, and here it is a factorial experiment (two factors, the strains of mice and the antibiotics treatment.

Any way if your experiment mono factor or factorial you can use analysis of variance to find all possible comparisons you wish to find

How many mice in each group? Depending on your n if it is less than 30 you can use Kruskal- Wallis ANOVA which is a non-parametric test. You can also take ideas from my mice papers. Your A and B groups are like my TA and TNA groups with a control group.

2-way-ANOVA, (2X3), (Group 2 levels X Treatment 3 levels), look for main effect for group of and main effect for treatment followed by post-comparison (adjusted), and also for interaction effect

In this research I must use Spss ....anova ....tuky analysis in the diference in the same group and student t test in difference between a and b groups ....