To add Patient Id as random effect or not in a linear mixed model I have the survey score for many patients for 4 intervals (in time). I wanted to see if there is significant change as time progresses (i.e. interval number goes from 1 to 4). I researched and found out a linear mixed effect model is the best way to analyze this. So if I have data like the following:
Patient 1
4 , 5 , 6 , 7
i.e. for each patient, just one score for each interval,
For the linear mixed model, If the patient id is in the "Subjects" field in SPSS, do I really have to include it as a random effect as well?
Here is the result without any random effect added to the model (just Interval is used as fixed effect):

P.S. Here is the result with random effect added:

Here is the summarized result for random and fixed:


 A: I understand that you have 77 patients who are measured 4 times each.
In this case, a mixed model is a good way to handle the repeated measures. You should fit random intercepts for patient ID. I can't comment about anything to do with spss, but the output should clearly say that it's a mixed effects model and it should estimate the variance for the random intercept, along with fixed effects for time and any other covariates. The estimate for time will answer your research question. If time is numeric / continuous then you will get an estimate for the linear slope, if it is categorical then you will get estimate for each time point.
I would also suggest plotting the data first in order to determine if a non-linear association is present.

Edit: Interpretation questions from comments after model output was posted into the question:

I can not understand what this output means, on one hand in the upper table it says that Intervals are overall significant, but then in the lower table it says that only Interval one has significant results ?

It appears that the interval is a categorical variable. The test in the upper table is an F test for the overall significance of that variable in the model. The individual tests are t tests, which test the hypothesis that each one is zero. It is not unusual for the overall F test to be significant while one or more of the individual levels are not.

Also interval 4 has been set to zero for some reason?

Interval 4 is set to zero because (and this is ony my assumption as I don't know anything about SPSS) "contrast coding" is used so in this case interval 4 is the "reference" level and the association of interval 4 with the outcome is included in the intercept. This is the default coding method in all the software that I know. The remaining three estimates are the contrast with the reference level - that is, the expected differece in the outcome variable between the reference level (interval 4) and the other 3. So:

*

*The expected value of the outcome is 11.49 at interval 4.

*There is a difference of -1.07 in the expected value of the outcome between interval 1 and inteval 4

*There is a difference of -0.23 in the expected value of the outcome between interval 2 and inteval 4

*There is a difference of -0.15 in the expected value of the outcome between interval 3 and inteval 4

*The overall trend is upwards but by far the biggest difference is between intervals 1 and 2 (0.84), between 2 and 3 it is 0.08 and between 3 and 4 it is 0.15.

You could change the reference level (for exxample to 1) but you would still get these results (but the p values would change)
Looking at the p values (Sig) in the output you could say:

*

*For the difference between the outcome at interval 1 and interval 4, the probability of obtaining these results, -1.07, (or results more extreme) is 0.002 if the true difference was actually zero.

*For the difference between the outcome at interval 2 and interval 4, the probability of obtaining these results, -0.23 (or results more extreme) is 0.535 if the true difference was actually zero.

*For the difference between the outcome at interval 3 and interval 4, the probability of obtaining these results, -0.23 (or results more extreme) is 0.607 if the true difference was actually zero.
So you can be confident that there is a negative association between intervals 1 and 4, but you are much less confident about what the values are in between.

Further edit to address new questions in the comments regarding other newly added output:

I just have one final question i swear, I have added the summarized result at the bottom, can you please let me know why there is no intercept shown for the random effect?

The row that says "Random Effects: patientid" in the model dimension table is the random intercept. The actual estimate for it is in the covariance parameters (2.11) , along with the measurement-level (unit-level) variance, from that you can compute "the intra-class correlation" (sometimes also called the "variance partition coeficient" which is often very useful. Different software adopts different ways of reporting things. Personally I dislike SPSS greatly, partly for reasons like this.

I do not know whether to look at "Estimates of fixed effects" table to get Sig. value for intervals or whether to look at "Covariance Parameters" table. And what do these individual sig. value mean for each interval?

It's in the fixed effects table. I think I explained what they mean in the previous edit. Each one is a t test of the hypothesis that the effects are zero. The smaller the p value (Sig) the smaller is the probaility that the actual data, or data more extreme, would be observed if the real/true value was zero. This is the definition of a p value (Sig in the SPSS output) but you must try to stop worrying about these. They should be banned by the statistics police ;)

I just want to get a conclusive result that shows whether results have improved over time, and indeed, during which intervals was this change most significant.

Again, I think I explained this in the last edit. There is strong evidence for a negative association between intervals 1 and 4, but very weak evidence for the points in between. You could try changing the reference level so that the results are contrasts with level 1 instead of level 4, and this might prove useful, however you should also be aware of running many tests (the "multiple testing problem"). Here is a link that talks about different coding schemes:
https://stats.idre.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/
Regarding your quest for a "conclusive result", unfortunately statistics doesn't really work like that. You should try to get a deeper understanding of what these kinds of tests mean. As an example, suppose you collect some data and test a null hypothesis about some association between 2 variables. First it is a mistake to think that you can infer anything to do with causation. Second, when your software computes a p value, suppose that p value is 0.04999. Some people would be very happy that they found a "significant result". On the other hand, if the p value was 0.05001 the sample people would be miserable. And yet, these results are the same, such a small difference in the p value could simply be the result of measuring 1 extra person (maybe there was 1 missing value because of an ink stain on a piece of paper). p values get smaller as the sample size gets bigger, so please try to understand what p values are, and how misleading and unhelpful they can be.
