What univariate analysis do you mean? If you can give some output (and if the data are confidential just a minimal working example), this will be easier to answer. Also, what software or are you doing the analyses by hand?

These tow tests are the same when you use same dependent and independent variables and only have one independent variable. But in univariate analysis, you should put the independent variable in fixed factor's box if you work with SPSS software. Also be careful abut the p-value you look, in univariate analysis there is a p-value for intercept.

What univariate analysis do you mean? If you can give some output (and if the data are confidential just a minimal working example), this will be easier to answer. Also, what software or are you doing the analyses by hand?

I completely agree with Daniel that the OP should be more clear about used methods, so I can only guess she is using OLS regression for univariate analysis.

Many thanks to all for your answers. Agree, my question was not that clear, a reflection perhaps of my state of confusion!  What I was trying to ask was if I perform a one way anova test using 1 independent variable and 1 dependent variable and then again another anova with another independent variable and same dependent variable, I obtain statistical significance levels of 0.02 and 0.06 respectively.  (I am using SPSS so the dependent variable is transferred to the dependent list and the independent variable to the factor list).  When I then perform a univariate analysis (two way anova) in SPSS (general linear model) using the same 2 independent variables and the same dependent variable, the statistical significance changes to 0.3 and 0.4 for the independent variables.

How associated are your 2 independent variables? I am not an SPSS user, but I assume the p values for the indep vars are their significance after controlling for the other. If they are associated, then the amount of extra variation accounted for by either of them may be small because each accounts for much of the same variation. This is where the instructor draws a Venn diagram.

This is a nice point; perhaps I did not understand your question very well. The analysis of variance for a one way classification is uivariate if you have only one dependent variable. You have a bivariate analysis when you have two dependent variables.

Let us suppose that you have one way ANOVA with a four treatments structure, which is a 2x2 factorial arrangement of factors A and B. It may be possible that the sum of squares of the A + B + AB effects; for orthogonal contrasts obtained is different from the sum of squares obtained by GLM for the treatment effect, due to the fact that have unequal number of observations per treatment.

Most software does ANOVA via matrix analysis, in order to simultaneously adjust all effects for the unequal number of observations in each cell. But even then, the results will be the same obtained even using conventional analysis of variance formulas by hand.

If you are talking about the results of performing individual “t” test between the treatments (which should not be done), without performing the ANOVA, of course you will find different results in the significance and probability levels . That's just because the estimator of the common variance in the ANOVA (mean square error), will be different to that used in each “t” test.

Remember that the variance analysis is an extension of “t” tests for comparing two independent samples.

For the one way ANOVA, the estimator of the common variance is:

MSE=[ (n1-1)Var1 + (n2-1)Var2+ . . .+ (nk-1)Vark ]/ [( n1 + n2+ . . . + nk)- k) ].

For each individual “t” test, the pooled variance is:

VARpooled=[ (n1-1)Var1 + (n2-1)Var2 ]/ [( n1 + n2)- 2) ].

" I perform a one way anova test using 1 independent variable and 1 dependent variable and then again another anova with another independent variable and same dependent variable, I obtain statistical significance levels of 0.02 and 0.06 respectively. (I am using SPSS so the dependent variable is transferred to the dependent list and the independent variable to the factor list). When I then perform a univariate analysis (two way anova) in SPSS (general linear model) using the same 2 independent variables and the same dependent variable, the statistical significance changes to 0.3 and 0.4 for the independent variables."

The short answer is yes it is possible.

Technically, two one-way ANOVAs with two different factors and one univariate GLM with two (fixed) factors will have the same total sum of squares. However, in ANOVA the total sum of squares are distributed into between and within groups, while in GLM they are distributed into each variables (including an intercept and an interaction between factors if you use it). Therefore, the degree of freedom will be different and may lead to different F-tests.

I prefer to see your problem in this way. Using ANOVA you are only interested to see the effect of an independent variable/factor to the dependent variable and 'ignore' or cannot control the effect of the other independent variable, while in GLM (two way ANOVA) you are interested with both variables in a single framework and can interpret a result of one ind variable by controlling the the other variable constant. When one of the ind variables is insignificant in GLM while each of them significant in ANOVA, you may interpret that one ind variable is more dominant in affecting the dep variable.

Dear Sarah

I think that you initially have a problem of two simple regression with one dependent and two different independent variables, and then a problem of multiple regression with the two variables incorporated at once. If the two independent variables are strongly correlated between them, then each may be a good predictor of the response, but when you try both at the same time, the SPSS is performing a Type 2 test (see for example The Four Types of Estimable Functions in http://www.math.wpi.edu/saspdf/stat/chap12.pdf) and then it is studying the contribution of each independent variable when the other is present. Then the p value may become very large for each, because each is trying to explain what was not explained by the other variable.

Let me present one example. Suppose that you have the weight (Y) of a high school sample of boys in pounds, and the height of each one of the boys in meters (X1) and in feets (X2), then you may find a very low p value for explaining the dependent weight variable when you use X1, and when you use X2, but when you use both at the same time the p value for each would be very large (and may be 1) because of the most strong correlation between the two independents variables. This is an extreme example because each independent variable is explaining the same relationship between the weight and the height and because between X1 and X2 there is only a change of scale.

In a problem like this you should use only one of the two independent variables.

Sarah, I agree with Guillermo 100%, I was trying to respond with something similar, but ended up copying and pasting the question and then posting that by mistake, I apologize for the confusion. Whenever you have highly correlated predictors, you don't have a unique 'best-fitting' model, and high VIF values lead to the type of results you are getting. 

It is possible that two test procedures intended for testing the same hypothesis on the same set of data (or sample) will produce different p-values (i.e. different levels of statistical significance). This is rather easy to understand from an example. Consider testing for a mean difference in a population before and after a treatment. Conventionally this would be tested by a one-sample t-test where the individual changes in the variable of interest are computed. Say that the value of the change was replaced by a + or a - depending on whether the change was positive or not. In this case the sign-test could be used for testing the same hypothesis on the same data. But since the sign-test ignore the information contained in the values it is less likely to detect a mean difference. Hence, the power of (detecting a false null-hypothesis) the sign-test is lower than the one-sample-test. In designing tests in statistics one strives for UMP (uniformly most powerful tests) - they are tests that best exploit the information contained in the data with relevance for the hypothesis. However most test routinely used in applied research are not UMP and consequently what you noted is unsurprising.  

How about doing a 2x2 ANOVA without the interaction term, if there truly isn't any? Wouldn't this be an adequate estimate of a factor's unique effect on DV because each controls for the other?