Both anova and multiple regression can be thought of as a form of general linear model †. For example, for either, you might use PROC GLM in SAS or lm in R.

So, anova and multiple regression can be exactly the same.

However, if you are using a different model for each, they will be different. Also, if you are sums of squares are calculated by different methods (Type I, Type II, or Type III), the results will be different.

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† Don't confuse this with generalized linear model.

How did you handle your IVs in regression? Assume they are categorical, did you specify them as categorical in regression?

Xiang Liu , yes, i treat my IVs as categorical in multiple regression.

Although the methods are closely related (cf Salvatore's answer) differences between anova and regression test results may arise because the H0/H1 alternative are not exactly the same (the degrees of freedom often differ). In anova you are looking for a difference between levels of IV, in regression for trends (monotonic if limited to linear effects). It may very well happen that a mild progression is undetected by anova and detected by regression. But with categorical IV, the values you give to the levels may affect the result(s).

In addition to Jean Maccario 's answer. For a DV (Y), with an IV with at least 3 categories (X1, X2, X3), the H0-hypothesis are

ANOVA's H0: \mu_{X1}=\mu_{X2}=\mu_{X3}

MV-Reg's H0: \beta_{X2}=0 and \beta_{X3}=0

which are obviously different. A very easy case to understand is when the categories X1 and X2 are the practically the same, but X3 not. So in this case, ANOVA will give insignificant result, while MV-Reg gives you a significant result for \beta_{X3}.

If you have significance in both of betas in MV-Reg, then your case is one of the very rare occassions, where the degree of freedom matters, and it helps to get the t-statistic across the alpha-level just barely. In this particular case, you should not reject the H0 hypothesis!

Even if your result shows significance, it does not mean that your null-hypothesis is false. It just mean that the probability of your data in this model is low.

Jean Maccario Christopher Paulus Imanto Thanks for your response. Are you saying that it is possible to have non-significant result in factorial ANOVA and significant result in multiple regression using the same variables?

yes - see below a little R demo

set.seed(1789)

x

I think this discussion is getting confused. Jean Maccario , the problem with your example is that the variables are not in fact the same. x is a numeric variable and xf is a categorical variable. Kavitha Selvaraja , the problem with your most recent question is that you asked about a significant result for anova vs. regression, but you didn't specify what hypothesis you are asking about. It would be best to give a specific example about what you are trying to ask about. Terms like "linking" and "contribute the most" in your original question are not specific enough for you to get a satisfactory answer. The best thing in these cases is to give a reproducible example (that is, make up some toy data!), and ask a specific question about the results. Otherwise, you are likely to get a lot of theoretical discussion that probably has nothing to do with what you are trying to ask. I suspect that is the current state of this thread.

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That being said, I'll try to clarify some confusion.

I think, the best way to think of these analyses is that they start with the same model. That is why for either anova or regression you might start with lm() in R or PROC GLM in SAS.

Once that model is fit, you can ask different hypotheses about the model.

I think this is the best way to think about these things. Don't start by thinking "We use anova for categorical IV, and regression for continuous IV". This is useful when you are first learning, or if you are doing calculations by hand, but it's not how these are approached in the age of using computers for analysis. Anova, regression, ancova, and so on, can all be conducted by testing hypotheses about a general linear model.

The important thing is to clarify in your mind what model you are using and what hypotheses you want to ask about that model.

To answer your question, Anova and regression are exactly the same if they use the same model. But if you ask different hypotheses about the model, of course you may get different answers.

If you are thinking about the model as a regression model, you might use summary(lm()) in R. This will tell you the coefficients for the model. You might want to know if these coefficients are significantly different from zero. So, you would conduct these hypothesis tests. These tests are also included in the summary(lm()) output, as is an r-squared value for the model.

Anova can be thought of as comparing two models. That is, the effect of A in the model Y ~ A + B would compare the model Y ~ A + B to the model Y ~ B. Here, you might use car:Anova(lm()) in R, and you want a p-value for the effect of A.

Note in this discussion that it doesn't matter what kind of variables A or B are. If A is a categorical variable, it will be dummy-coded and entered into the model as a series of dichotomous variables, and the model coefficients will refer to the dummy variables. But the anova results will refer to the collection of dummy variables that constitute A. Again, it just depends on what hypothesis you are asking about the model.

I did the two tests, the first with X as continuous, the second with X as categorical.

see anova table below paying attention to the df's and the IV's involved

the command

anova(lm(yobs1~x))

gives (forgive the poor format)

Response: yobs1

Df Sum Sq Mean Sq F value Pr(>F)

x 1 7.2636 7.2636 4.97 0.03877 *

Residuals 18 26.3068 1.4615

the command

anova(lm(yobs1~xf)) ## note IV is "xf" and not "x")

gives

Response: yobs1

Df Sum Sq Mean Sq F value Pr(>F)

xf 3 8.3884 2.7961 1.7766 0.1921

Residuals 16 25.1820 1.5739

I do agree that the models are different, but I maintain that, with the same data, you may have a p-value < 0.05 in one case, and > 0.05 in the other, not necessarily in the order above.

I do agree also that giving "toy data" may always be useful.

Kavitha Selvaraja, yes it is possible to have different significance result from different methods.

Given H_0 is false: as stated above, the hypotheses are different, therefore it is possible to have different result. In any case, the proof for statistical methods are mostly "only" true asymptotically (or sometime "almost surely" true). So as long as the population is finite, different result is not surprising at all, but of course rare.

Given H_0 is true: there is no guarantee, that a true H_0 would give you a "high" p-value. In fact, given a true H_0 and a random data, the p-value should be uniformly distributed. So again: different significance result is common to see, even if the H_0 of both methods are exactly the same.

Salvatore S. Mangiafico Thanks for the explanation. It is very comprehensive. Appreciate it.

Yes, I am using the same model with different hypothesis. My independent variables are entrepreneurial leadership, school culture and school innovativeness while my dependent variable is school performance.

My hypothesis is:

1. There is a significant effect between entrepreneurial leadership, school culture and school innovativeness towards school performance? 

2. There is a significant interaction effect between entrepreneurial leadership, school culture, school innovativeness and type of schools towards school performance? 

My Research Question is:

Which predictor (independent variables) contributes most to the performance of schools?

I run Factorial ANOVA to identify the main effect and interaction effect between the variables (Refer to my hypothesis). The result shows that there is no significant effect between variables.

I run multiple regression to answer my research question. The result shows that school innovativeness contributes most to school performance.

Therefore, I would like to get your opinion if my result makes sense. Kindly advise.

It probably makes sense, but, to clarify:

What statistic are you using to determine that innovativeness contributes the most to school performance?

What kind of variables are your IV's? Are they categorical? Continuous?

Do I understand that none of the variables or interactions were significant in the anova?

In ANOVA we always look for difference between groups in independent variables where as in multiple regression we always look for relationship and for prediction. Significant factors associated with outcome can be identified and model fitness tells us whether model is good enough for prediction.