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

Question

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?

Answer 1

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
-------------+------------------------------           Adj R-squared =  0.3963
       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
-------------+------------------------------           Adj R-squared =  0.4085
       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 regresion 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).

Answer 2

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.

Answer 3

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.

Answer 4

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?

Answer 5

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.

Answer 6

One way to understand this is the geometry of least squares as @StasK suggests.

Another is to realize it means that X is related to Y when controlling for the other variables, but not alone. You say X relates to unique variance in Y. This is right. The unique variance in Y, though, is different from the total variance. So, what variance are the other variables removing?

It would help if you could tell us your variables.