Here are regression analyses between nino34 and link strength. The code and output snippets are in R. This is on the full 1950-2013 data.

+ Link strength vs nino34, no lag
+ although statistically significant (p=.0002) the correlation only explains about 1.6% of the variance.

Call:

Residuals:
Min 1Q Median 3Q Max
-1.9899 -0.5747 -0.0789 0.5044 2.4438

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.005587 0.029435 0.190 0.849523
d\$link 0.350309 0.094849 3.693 0.000237 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8157 on 766 degrees of freedom
Multiple R-squared: 0.0175, Adjusted R-squared: 0.01621
F-statistic: 13.64 on 1 and 766 DF, p-value: 0.000237

+ Link strength vs nino34, with 6 month lag
+ statistically significant (p < 2.2e-16) and explains about 8.7% of the variance.

Call:
lm(formula = d\$ANOM[7:nrow(d)] ~ d\$link[1:(nrow(d) - 6)])

Residuals:
Min 1Q Median 3Q Max
-2.01307 -0.54718 -0.03153 0.48593 2.33848

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.01621 0.02834 0.572 0.567
d\$link[1:(nrow(d) - 6)] 0.78163 0.09113 8.577 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7822 on 760 degrees of freedom
Multiple R-squared: 0.08826, Adjusted R-squared: 0.08706
F-statistic: 73.57 on 1 and 760 DF, p-value: < 2.2e-161.6

+ Link nino34 vs nino34, with 6 month lag
+ statistically significant (p<2.2e-16) and explains about 16.1% of the variance.

> summary(lm(d\$ANOM[7:nrow(d)] ~ d\$ANOM[1:(nrow(d)-6)]))

Call:
lm(formula = d\$ANOM[7:nrow(d)] ~ d\$ANOM[1:(nrow(d) - 6)])

Residuals:
Min 1Q Median 3Q Max
-1.92681 -0.45092 -0.03014 0.45603 2.25653

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.01162 0.02717 0.428 0.669
d\$ANOM[1:(nrow(d) - 6)] 0.39918 0.03295 12.115 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.75 on 760 degrees of freedom
Multiple R-squared: 0.1619, Adjusted R-squared: 0.1608
F-statistic: 146.8 on 1 and 760 DF, p-value: < 2.2e-16