# How can a regression be significant yet all predictors be non-significant?

My multiple regression analysis model has a statistically significant F value however all beta values are statistically non-significant.

All the regression assumptions are met. No multicollinearity was found. Correlations among all predictors are all less than 0.60.

What else might be the cause of the insignificant predictors?

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Is the constant also insignificant? How many cases are involved? How many variables? –  whuber Aug 19 '11 at 5:34
How was multicollinearity diagnosed? There are many methods, some are more informative than others. The more you tell us, the better the community can answer. –  StasK Aug 19 '11 at 14:43
This question has become a FAQ. Some of the answers here were merged from substantially similar threads. –  whuber Aug 7 '12 at 15:50
See also here: f-and-t-statistics-in-a-regression, & for a discussion of the opposite case, see here: significant t-test vs non-significant F-statistic. –  gung Sep 13 '12 at 15:12

It takes very little correlation among the independent variables to cause this.

To see why, try the following:

• Draw 50 sets of ten vectors $(x_1, x_2, \ldots, x_{10})$ with coefficients iid standard normal.

• Compute $y_i = (x_i + x_{i+1})/\sqrt{2}$ for $i = 1, 2, \ldots, 9$. This makes the $y_i$ individually standard normal but with some correlations among them.

• Compute $w = x_1 + x_2 + \cdots + x_{10}$. Note that $w = \sqrt{2}(y_1 + y_3 + y_5 + y_7 + y_9)$.

• Add some independent normally distributed error to $w$. With a little experimentation I found that $z = w + \varepsilon$ with $\varepsilon \sim N(0, 6)$ works pretty well. Thus, $z$ is the sum of the $x_i$ plus some error. It is also the sum of some of the $y_i$ plus the same error.

We will consider the $y_i$ to be the independent variables and $z$ the dependent variable.

Here's a scatterplot matrix of one such dataset, with $z$ along the top and left and the $y_i$ proceeding in order.

The expected correlations among $y_i$ and $y_j$ are $1/2$ when $|i-j|=1$ and $0$ otherwise. The realized correlations range up to 62%. They show up as tighter scatterplots next to the diagonal.

Look at the regression of $z$ against the $y_i$:

      Source |       SS       df       MS              Number of obs =      50
-------------+------------------------------           F(  9,    40) =    4.57
Model |  1684.15999     9  187.128887           Prob > F      =  0.0003
Residual |  1636.70545    40  40.9176363           R-squared     =  0.5071
Total |  3320.86544    49  67.7727641           Root MSE      =  6.3967

------------------------------------------------------------------------------
z |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y1 |   2.184007   1.264074     1.73   0.092    -.3707815    4.738795
y2 |   1.537829   1.809436     0.85   0.400    -2.119178    5.194837
y3 |   2.621185   2.140416     1.22   0.228    -1.704757    6.947127
y4 |   .6024704   2.176045     0.28   0.783    -3.795481    5.000421
y5 |   1.692758   2.196725     0.77   0.445    -2.746989    6.132506
y6 |   .0290429   2.094395     0.01   0.989    -4.203888    4.261974
y7 |   .7794273   2.197227     0.35   0.725    -3.661333    5.220188
y8 |  -2.485206    2.19327    -1.13   0.264     -6.91797    1.947558
y9 |   1.844671   1.744538     1.06   0.297    -1.681172    5.370514
_cons |   .8498024   .9613522     0.88   0.382    -1.093163    2.792768
------------------------------------------------------------------------------


The F statistic is highly significant but none of the independent variables is, even without any adjustment for all 9 of them.

To see what's going on, consider the regression of $z$ against just the odd-numbered $y_i$:

      Source |       SS       df       MS              Number of obs =      50
-------------+------------------------------           F(  5,    44) =    7.77
Model |  1556.88498     5  311.376997           Prob > F      =  0.0000
Residual |  1763.98046    44  40.0904649           R-squared     =  0.4688
Total |  3320.86544    49  67.7727641           Root MSE      =  6.3317

------------------------------------------------------------------------------
z |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y1 |   2.943948   .8138525     3.62   0.001     1.303736     4.58416
y3 |   3.403871   1.080173     3.15   0.003     1.226925    5.580818
y5 |   2.458887    .955118     2.57   0.013      .533973    4.383801
y7 |  -.3859711   .9742503    -0.40   0.694    -2.349443    1.577501
y9 |   .1298614   .9795983     0.13   0.895    -1.844389    2.104112
_cons |   1.118512   .9241601     1.21   0.233    -.7440107    2.981034
------------------------------------------------------------------------------


Some of these variables are highly significant, even with a Bonferroni adjustment. (There's much more that can be said by looking at these results, but it would take us away from the main point.)

The intuition behind this is that $z$ depends primarily on a subset of the variables (but not necessarily on a unique subset). The complement of this subset ($y_2, y_4, y_6, y_8$) adds essentially no information about $z$ due to correlations—however slight—with the subset itself.

This sort of situation will arise in time series analysis. We can consider the subscripts to be times. The construction of the $y_i$ has induced a short-range serial correlation among them, much like many time series. Due to this, we lose little information by subsampling the series at regular intervals.

One conclusion we can draw from this is that when too many variables are included in a model they can mask the truly significant ones. The first sign of this is the highly significant overall F statistic accompanied by not-so-significant t-tests for the individual coefficients. (Even when some of the variables are individually significant, this does not automatically mean the others are not. That's one of the basic defects of stepwise regression strategies: they fall victim to this masking problem.) Incidentally, the variance inflation factors in the first regression range from 2.55 to 6.09 with a mean of 4.79: just on the borderline of diagnosing some multicollinearity according to the most conservative rules of thumb; well below the threshold according to other rules (where 10 is an upper cutoff).

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Great answer. A plus 1 from me. I would have liked to give it more. –  Michael Chernick May 4 '12 at 22:11

### Multicollinearity

• As you note, and as has been discussed in this previous question, high levels of multicollinearity is one major cause of a statistically significant $R^2$ but statically non-significant predictors.
• Of course, multicollinearity is not just about an absolute threshold. Standard errors on regression coefficients will increase as intercorrelations with the focal predictor increase.

### Multiple almost significant predictors

• Even if you had no multicollinearity, you can still get non-significant predictors and an overall significant model if two or more individual predictors are close to significant and thus collectively, the overall prediction passes the threshold of statistical significance. For example, using an alpha of .05, if you had two predictors with p-values of .06, and .07, then I wouldn't be surprised if the overall model had a p<.05.
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The answer you get depends on the question you ask. In addition to the points already made, the individual parameters F values and the overall model F values answer different questions, so they get different answers. I have seen this happen even when the individual F values are not that close to significant, especially if the model has more than 2 or 3 IVs. I do not know of any way to combine the individual p-values and get anything meaningful, althought there may be a way.

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(-1) Yes - the original poster is noting that he/she has seen it happen too. The question was what exactly are some things that may cause this other than collinearity and I don't see how this is an answer. –  Macro Aug 19 '11 at 10:46
@Macro The downvote seems a little harsh, because there is a useful and valid observation in this reply: the tests for overall significance and for individual variable significance "answer different questions." Admittedly that's qualitative, but no more so then the first answer with many upvotes; and to that answer it adds some valid intuition, arguably making it an improvement over that answer. –  whuber Aug 19 '11 at 14:18
I never said there wasn't valid information or intuition supplied by this reply. If I had a good answer for this question I'd have responded by now - it's kind of a tough question - I was only saying that this response doesn't seem to answer the question in any sense of the word. –  Macro Aug 19 '11 at 14:27

One other thing to keep in mind is that the tests on the individual coefficients each assume that all of the other predictors are in the model. In other words each predictor is not significant as long as all of the other predictors are in the model. There must be some interaction or interdependence between two or more of your predictors.

As someone else asked above - how did you diagnose a lack of multicollinearity?

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Consider the following model: $X_1 \sim N(0,1)$, $X_2 = a X_1 + \delta$, $Y = bX_1 + cX_2 + \epsilon$, $\delta$, $\epsilon$ and $X_1$ are all mutually independent $N(0,1)$.

Then $${\rm Cov}(X_2,Y) = {\rm E}[(aX_1+\delta)(bX_1+cX_2+\epsilon)]={\rm E}[(aX_1+\delta)(\{b+ac\}X_1+c\delta+\epsilon)]=a(b+ac)+c$$

We can set this to zero with say $a=1$, $b=2$ and $c=-1$. Yet all the relations will obviously be there and easily detectable with regression analysis.

You said that you understand the issue of variables being correlated and regression being insignificant better; it probably means that you have been conditioned by frequent mentioning of multicollinearity, but you would need to boost your understanding of the geometry of least squares.

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