# one-sided z statistic

Hello, I am estimating some non-linear models and I need to estimate the corresponding one-sided z-statistic for each regressor and a two sided z-statistic for the intercept. Could you please provide some guidance for their computation, as well as their p-values? Thank you, Karla

1

As an example, consider the non-linear regression results below, obtained using simulated data and the Maximum Likelihood MT application module:
As an example, consider the non-linear regression results below, obtained using simulated data and the Maximum Likelihood MT application module:

```Log-likelihood        -208.823
Number of cases     200

Covariance of the parameters computed by the following method:
ML covariance matrix
Parameters    Estimates     Std. err.  Est./s.e.  Prob.    Gradient
---------------------------------------------------------------------
b0[1,1]    0.5485        0.2488       2.205   0.0275      0.0000
b[1,1]     0.0693        0.0567       1.223   0.2214      0.0002
```

In this example, the probabilities listed in the table both correspond to two-sided tests of the hypothesis that b0[1,1]≠0 and that b[1,1]≠0. We can find both the two-side p-values in table and one-sided p-values using GAUSS procedure cdfN. This procedure computes integral of Normal distribution: lower tail, or cdf for a given z-statistic. Hence, to compute the hypothesis test that a coefficient is different from a null hypothesis, h0, we first need to find the z-statistic:

```//Null hypothesis
h0=0;

//Z_Stat = (estimate - h0)/se
//Find intercept Z-statistic
Z_b0 = (0.5485 - h0)/0.2488;

//Find b1 Z-statistic
Z_b1 = (0.0693 - h0)/0.0567;
```

Next, to find the probabilities corresponding to the one-sided test:

```//Find one-sided p-value Z_b0
p_onesided_b0 = 1-cdFN(z_b0);

//Find one-sided p-value Z_b1
p_onesided_b1 = 1-cdFN(z_b1);
```

Finally, to find the two-sided p-values (listed in the table), we multiply the one-sided by values by 2:

```//Find two-sided p-value Z_b0
p_twosided_b0 = 2*p_onesided_b0;

//Find two-sided p-value Z_b1
p_twosided_b1 = 2*p_onesided_b1;
```

aptech

1,728

0

Thank you very much for answering my question. Could you please also provide a way of computing the one-sided critical values |z| for significance levels 99%, 95% and 90%, assuming a sample size of 700 observations? Thanks again.

0

To find the one-sided critical value for a normally distributed variable, let's start from the equations in my original response for finding the p-value for an associated z-statistic:

```//Find one-sided p-value Z_b0
p_onesided_b0 = 1-cdfn(z_b0);
```

If we rearrange this equation we get:

```//Find two-sided p-value Z_b0
cdfn(z_b0) = 1 - p_onesided_b0;
```

For the 99%, 95%, and 90% significance levels, the associated probability values, α, are equal to 1%, 5%, and 10%. Note, additionally, that α is equivalent to the variable p_onesided_b0. So if we use 99% as an example:

```//Find two-sided p-value Z_b0
cdfn(z_b0) = 1 - 0.01 = 0.99;
```

To solve this for the associated critical values we can use the GAUSS function cdFni. This function is the inverse of cdFN. To find the critical value for a 99% significance level:

```//Find two-sided p-value Z_b0
z_b0 = cdfni(0.99);
```

We can also easily find this critical value using the GAUSS command line by typing cdFNi(0.99) and pressing enter:

```>>cdfni(0.99)
2.3263479

>>cdfni(0.95)
1.6448536

>>cdfni(0.90)
1.2815516
```

Eric

90

1

As an example, consider the non-linear regression results below, obtained using simulated data and the Maximum Likelihood MT application module:
As an example, consider the non-linear regression results below, obtained using simulated data and the Maximum Likelihood MT application module:

```Log-likelihood        -208.823
Number of cases     200

Covariance of the parameters computed by the following method:
ML covariance matrix
Parameters    Estimates     Std. err.  Est./s.e.  Prob.    Gradient
---------------------------------------------------------------------
b0[1,1]    0.5485        0.2488       2.205   0.0275      0.0000
b[1,1]     0.0693        0.0567       1.223   0.2214      0.0002
```

In this example, the probabilities listed in the table both correspond to two-sided tests of the hypothesis that b0[1,1]≠0 and that b[1,1]≠0. We can find both the two-side p-values in table and one-sided p-values using GAUSS procedure cdfN. This procedure computes integral of Normal distribution: lower tail, or cdf for a given z-statistic. Hence, to compute the hypothesis test that a coefficient is different from a null hypothesis, h0, we first need to find the z-statistic:

```//Null hypothesis
h0=0;

//Z_Stat = (estimate - h0)/se
//Find intercept Z-statistic
Z_b0 = (0.5485 - h0)/0.2488;

//Find b1 Z-statistic
Z_b1 = (0.0693 - h0)/0.0567;
```

Next, to find the probabilities corresponding to the one-sided test:

```//Find one-sided p-value Z_b0
p_onesided_b0 = 1-cdFN(z_b0);

//Find one-sided p-value Z_b1
p_onesided_b1 = 1-cdFN(z_b1);
```

Finally, to find the two-sided p-values (listed in the table), we multiply the one-sided by values by 2:

```//Find two-sided p-value Z_b0
p_twosided_b0 = 2*p_onesided_b0;

//Find two-sided p-value Z_b1
p_twosided_b1 = 2*p_onesided_b1;
```
0

Thank you very much for answering my question. Could you please also provide a way of computing the one-sided critical values |z| for significance levels 99%, 95% and 90%, assuming a sample size of 700 observations? Thanks again.

0

To find the one-sided critical value for a normally distributed variable, let's start from the equations in my original response for finding the p-value for an associated z-statistic:

```//Find one-sided p-value Z_b0
p_onesided_b0 = 1-cdfn(z_b0);
```

If we rearrange this equation we get:

```//Find two-sided p-value Z_b0
cdfn(z_b0) = 1 - p_onesided_b0;
```

For the 99%, 95%, and 90% significance levels, the associated probability values, α, are equal to 1%, 5%, and 10%. Note, additionally, that α is equivalent to the variable p_onesided_b0. So if we use 99% as an example:

```//Find two-sided p-value Z_b0
cdfn(z_b0) = 1 - 0.01 = 0.99;
```

To solve this for the associated critical values we can use the GAUSS function cdFni. This function is the inverse of cdFN. To find the critical value for a 99% significance level:

```//Find two-sided p-value Z_b0
z_b0 = cdfni(0.99);
```

We can also easily find this critical value using the GAUSS command line by typing cdFNi(0.99) and pressing enter:

```>>cdfni(0.99)
2.3263479

>>cdfni(0.95)
1.6448536

>>cdfni(0.90)
1.2815516
```

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